Quadratic roots, narrow range

Percentage Accurate: 55.1% → 91.6%

Time: 15.7s

Alternatives: 14

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ t_1 := \mathsf{fma}\left(2, t\_0, b\right)\\ t_2 := b + t\_0 \cdot -2\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, t\_2, -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{t\_1}, \sqrt{t\_2}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{4} \cdot \left(a \cdot \left(-5 \cdot \frac{a}{{b}^{7}} - \frac{2}{c \cdot {b}^{5}}\right)\right) - {c}^{2} \cdot {b}^{-3}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))) (t_1 (fma 2.0 t_0 b)) (t_2 (+ b (* t_0 -2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -2000.0)
     (/
      (/ (fma t_1 t_2 (- (pow b 2.0))) (fma (sqrt t_1) (sqrt t_2) b))
      (* a 2.0))
     (-
      (*
       a
       (-
        (*
         (pow c 4.0)
         (* a (- (* -5.0 (/ a (pow b 7.0))) (/ 2.0 (* c (pow b 5.0))))))
        (* (pow c 2.0) (pow b -3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double t_1 = fma(2.0, t_0, b);
	double t_2 = b + (t_0 * -2.0);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -2000.0) {
		tmp = (fma(t_1, t_2, -pow(b, 2.0)) / fma(sqrt(t_1), sqrt(t_2), b)) / (a * 2.0);
	} else {
		tmp = (a * ((pow(c, 4.0) * (a * ((-5.0 * (a / pow(b, 7.0))) - (2.0 / (c * pow(b, 5.0)))))) - (pow(c, 2.0) * pow(b, -3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	t_1 = fma(2.0, t_0, b)
	t_2 = Float64(b + Float64(t_0 * -2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -2000.0)
		tmp = Float64(Float64(fma(t_1, t_2, Float64(-(b ^ 2.0))) / fma(sqrt(t_1), sqrt(t_2), b)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64((c ^ 4.0) * Float64(a * Float64(Float64(-5.0 * Float64(a / (b ^ 7.0))) - Float64(2.0 / Float64(c * (b ^ 5.0)))))) - Float64((c ^ 2.0) * (b ^ -3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * t$95$0 + b), $MachinePrecision]}, Block[{t$95$2 = N[(b + N[(t$95$0 * -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], -2000.0], N[(N[(N[(t$95$1 * t$95$2 + (-N[Power[b, 2.0], $MachinePrecision])), $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(a * N[(N[(-5.0 * N[(a / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(c * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2e3

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

   3. Simplified 94.3%

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

      1. *-commutative 94.3%

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

      2. metadata-eval 94.3%

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

      3. distribute-lft-neg-in 94.3%

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

      4. distribute-rgt-neg-in 94.3%

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

      5. *-commutative 94.3%

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

      6. fmm-def 94.1%

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

      7. add-sqr-sqrt 94.1%

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

      8. difference-of-squares 94.3%

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

      9. associate-*l* 94.3%

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

      10. sqrt-prod 94.3%

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

      11. metadata-eval 94.3%

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

      12. associate-*l* 94.3%

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

      13. sqrt-prod 94.3%

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

      14. metadata-eval 94.3%

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

   6. Applied egg-rr 94.3%

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

      1. flip-- 93.8%

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

      2. add-sqr-sqrt 94.1%

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

      3. fmm-def 94.9%

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

      4. +-commutative 94.9%

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

      5. fma-define 94.9%

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

      6. cancel-sign-sub-inv 94.9%

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

      7. metadata-eval 94.9%

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

      8. pow2 94.9%

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

   8. Applied egg-rr 95.0%

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

   if -2e3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 53.6%

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

      1. *-commutative 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in a around 0 93.7%

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

   6. Taylor expanded in c around inf 93.7%

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

   7. Taylor expanded in a around 0 93.7%

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

      1. associate-*r/ 93.7%

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

      2. metadata-eval 93.7%

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

      3. *-commutative 93.7%

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

   9. Simplified 93.7%

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

       1. pow1 93.7%

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

       2. mul-1-neg 93.7%

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

       3. div-inv 93.7%

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

       4. pow-flip 93.7%

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

       5. metadata-eval 93.7%

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

   11. Applied egg-rr 93.7%

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

       1. unpow1 93.7%

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

       2. distribute-rgt-neg-in 93.7%

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

   13. Simplified 93.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right), b + \sqrt{a \cdot c} \cdot -2, -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right)}, \sqrt{b + \sqrt{a \cdot c} \cdot -2}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{4} \cdot \left(a \cdot \left(-5 \cdot \frac{a}{{b}^{7}} - \frac{2}{c \cdot {b}^{5}}\right)\right) - {c}^{2} \cdot {b}^{-3}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, t\_0, -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{a}, \sqrt{t\_0}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{4} \cdot \left(a \cdot \left(-5 \cdot \frac{a}{{b}^{7}} - \frac{2}{c \cdot {b}^{5}}\right)\right) - {c}^{2} \cdot {b}^{-3}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c -4.0 (/ (pow b 2.0) a))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -2000.0)
     (/ (/ (fma a t_0 (- (pow b 2.0))) (fma (sqrt a) (sqrt t_0) b)) (* a 2.0))
     (-
      (*
       a
       (-
        (*
         (pow c 4.0)
         (* a (- (* -5.0 (/ a (pow b 7.0))) (/ 2.0 (* c (pow b 5.0))))))
        (* (pow c 2.0) (pow b -3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, -4.0, (pow(b, 2.0) / a));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -2000.0) {
		tmp = (fma(a, t_0, -pow(b, 2.0)) / fma(sqrt(a), sqrt(t_0), b)) / (a * 2.0);
	} else {
		tmp = (a * ((pow(c, 4.0) * (a * ((-5.0 * (a / pow(b, 7.0))) - (2.0 / (c * pow(b, 5.0)))))) - (pow(c, 2.0) * pow(b, -3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, -4.0, Float64((b ^ 2.0) / a))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -2000.0)
		tmp = Float64(Float64(fma(a, t_0, Float64(-(b ^ 2.0))) / fma(sqrt(a), sqrt(t_0), b)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64((c ^ 4.0) * Float64(a * Float64(Float64(-5.0 * Float64(a / (b ^ 7.0))) - Float64(2.0 / Float64(c * (b ^ 5.0)))))) - Float64((c ^ 2.0) * (b ^ -3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * -4.0 + N[(N[Power[b, 2.0], $MachinePrecision] / a), $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], -2000.0], N[(N[(N[(a * t$95$0 + (-N[Power[b, 2.0], $MachinePrecision])), $MachinePrecision] / N[(N[Sqrt[a], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(a * N[(N[(-5.0 * N[(a / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(c * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2e3

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in a around inf 94.1%

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

      1. flip-- 93.8%

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

      2. add-sqr-sqrt 94.6%

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

      3. unpow2 94.6%

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

      4. fmm-def 95.0%

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

      5. *-commutative 95.0%

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

      6. fma-define 95.0%

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

      7. sqrt-prod 94.8%

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

      8. fma-define 95.0%

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

      9. *-commutative 95.0%

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

      10. fma-define 95.0%

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

   7. Applied egg-rr 95.0%

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

   if -2e3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 53.6%

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

      1. *-commutative 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in a around 0 93.7%

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

   6. Taylor expanded in c around inf 93.7%

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

   7. Taylor expanded in a around 0 93.7%

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

      1. associate-*r/ 93.7%

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

      2. metadata-eval 93.7%

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

      3. *-commutative 93.7%

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

   9. Simplified 93.7%

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

       1. pow1 93.7%

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

       2. mul-1-neg 93.7%

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

       3. div-inv 93.7%

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

       4. pow-flip 93.7%

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

       5. metadata-eval 93.7%

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

   11. Applied egg-rr 93.7%

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

       1. unpow1 93.7%

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

       2. distribute-rgt-neg-in 93.7%

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

   13. Simplified 93.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right), -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{a}, \sqrt{\mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{4} \cdot \left(a \cdot \left(-5 \cdot \frac{a}{{b}^{7}} - \frac{2}{c \cdot {b}^{5}}\right)\right) - {c}^{2} \cdot {b}^{-3}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2000:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(2, t\_0, b\right) \cdot \left(b + t\_0 \cdot -2\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{4} \cdot \left(a \cdot \left(-5 \cdot \frac{a}{{b}^{7}} - \frac{2}{c \cdot {b}^{5}}\right)\right) - {c}^{2} \cdot {b}^{-3}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -2000.0)
     (/ 1.0 (/ (* a 2.0) (- (sqrt (* (fma 2.0 t_0 b) (+ b (* t_0 -2.0)))) b)))
     (-
      (*
       a
       (-
        (*
         (pow c 4.0)
         (* a (- (* -5.0 (/ a (pow b 7.0))) (/ 2.0 (* c (pow b 5.0))))))
        (* (pow c 2.0) (pow b -3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -2000.0) {
		tmp = 1.0 / ((a * 2.0) / (sqrt((fma(2.0, t_0, b) * (b + (t_0 * -2.0)))) - b));
	} else {
		tmp = (a * ((pow(c, 4.0) * (a * ((-5.0 * (a / pow(b, 7.0))) - (2.0 / (c * pow(b, 5.0)))))) - (pow(c, 2.0) * pow(b, -3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -2000.0)
		tmp = Float64(1.0 / Float64(Float64(a * 2.0) / Float64(sqrt(Float64(fma(2.0, t_0, b) * Float64(b + Float64(t_0 * -2.0)))) - b)));
	else
		tmp = Float64(Float64(a * Float64(Float64((c ^ 4.0) * Float64(a * Float64(Float64(-5.0 * Float64(a / (b ^ 7.0))) - Float64(2.0 / Float64(c * (b ^ 5.0)))))) - Float64((c ^ 2.0) * (b ^ -3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2000.0], N[(1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(2.0 * t$95$0 + b), $MachinePrecision] * N[(b + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(a * N[(N[(-5.0 * N[(a / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(c * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2e3

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

   3. Simplified 94.3%

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

      1. *-commutative 94.3%

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

      2. metadata-eval 94.3%

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

      3. distribute-lft-neg-in 94.3%

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

      4. distribute-rgt-neg-in 94.3%

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

      5. *-commutative 94.3%

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

      6. fmm-def 94.1%

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

      7. add-sqr-sqrt 94.1%

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

      8. difference-of-squares 94.3%

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

      9. associate-*l* 94.3%

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

      10. sqrt-prod 94.3%

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

      11. metadata-eval 94.3%

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

      12. associate-*l* 94.3%

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

      13. sqrt-prod 94.3%

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

      14. metadata-eval 94.3%

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

   6. Applied egg-rr 94.3%

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

      1. clear-num 94.4%

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

      2. inv-pow 94.4%

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

      3. +-commutative 94.4%

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

      4. fma-define 94.4%

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

      5. cancel-sign-sub-inv 94.4%

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

      6. metadata-eval 94.4%

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

   8. Applied egg-rr 94.4%

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

      1. unpow-1 94.4%

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

   10. Simplified 94.4%

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

   if -2e3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 53.6%

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

      1. *-commutative 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in a around 0 93.7%

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

   6. Taylor expanded in c around inf 93.7%

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

   7. Taylor expanded in a around 0 93.7%

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

      1. associate-*r/ 93.7%

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

      2. metadata-eval 93.7%

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

      3. *-commutative 93.7%

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

   9. Simplified 93.7%

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

       1. pow1 93.7%

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

       2. mul-1-neg 93.7%

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

       3. div-inv 93.7%

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

       4. pow-flip 93.7%

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

       5. metadata-eval 93.7%

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

   11. Applied egg-rr 93.7%

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

       1. unpow1 93.7%

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

       2. distribute-rgt-neg-in 93.7%

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

   13. Simplified 93.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2000:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \left(b + \sqrt{a \cdot c} \cdot -2\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{4} \cdot \left(a \cdot \left(-5 \cdot \frac{a}{{b}^{7}} - \frac{2}{c \cdot {b}^{5}}\right)\right) - {c}^{2} \cdot {b}^{-3}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(2, t\_0, b\right) \cdot \left(b + t\_0 \cdot -2\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.1)
     (/ 1.0 (/ (* a 2.0) (- (sqrt (* (fma 2.0 t_0 b) (+ b (* t_0 -2.0)))) b)))
     (-
      (*
       a
       (*
        (pow c 2.0)
        (+ (* -2.0 (/ (* a c) (pow b 5.0))) (/ -1.0 (pow b 3.0)))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.1) {
		tmp = 1.0 / ((a * 2.0) / (sqrt((fma(2.0, t_0, b) * (b + (t_0 * -2.0)))) - b));
	} else {
		tmp = (a * (pow(c, 2.0) * ((-2.0 * ((a * c) / pow(b, 5.0))) + (-1.0 / pow(b, 3.0))))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.1)
		tmp = Float64(1.0 / Float64(Float64(a * 2.0) / Float64(sqrt(Float64(fma(2.0, t_0, b) * Float64(b + Float64(t_0 * -2.0)))) - b)));
	else
		tmp = Float64(Float64(a * Float64((c ^ 2.0) * Float64(Float64(-2.0 * Float64(Float64(a * c) / (b ^ 5.0))) + Float64(-1.0 / (b ^ 3.0))))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.1], N[(1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(2.0 * t$95$0 + b), $MachinePrecision] * N[(b + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.10000000000000001

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

   3. Simplified 80.8%

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

      1. *-commutative 80.8%

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

      2. metadata-eval 80.8%

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

      3. distribute-lft-neg-in 80.8%

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

      4. distribute-rgt-neg-in 80.8%

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

      5. *-commutative 80.8%

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

      6. fmm-def 80.7%

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

      7. add-sqr-sqrt 80.7%

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

      8. difference-of-squares 80.8%

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

      9. associate-*l* 80.8%

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

      10. sqrt-prod 80.8%

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

      11. metadata-eval 80.8%

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

      12. associate-*l* 80.8%

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

      13. sqrt-prod 80.8%

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

      14. metadata-eval 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. clear-num 80.8%

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

      2. inv-pow 80.8%

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

      3. +-commutative 80.8%

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

      4. fma-define 80.8%

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

      5. cancel-sign-sub-inv 80.8%

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

      6. metadata-eval 80.8%

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

   8. Applied egg-rr 80.8%

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

      1. unpow-1 80.8%

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

   10. Simplified 80.8%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.7%

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

      1. *-commutative 48.7%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 96.2%

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

   6. Taylor expanded in c around inf 96.2%

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

   7. Taylor expanded in c around 0 94.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) \cdot \left(b + \sqrt{a \cdot c} \cdot -2\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.1)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (*
     a
     (* (pow c 2.0) (+ (* -2.0 (/ (* a c) (pow b 5.0))) (/ -1.0 (pow b 3.0)))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.1) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (a * (pow(c, 2.0) * ((-2.0 * ((a * c) / pow(b, 5.0))) + (-1.0 / pow(b, 3.0))))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64((c ^ 2.0) * Float64(Float64(-2.0 * Float64(Float64(a * c) / (b ^ 5.0))) + Float64(-1.0 / (b ^ 3.0))))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.1], 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[(a * N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.10000000000000001

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

   3. Simplified 80.8%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.7%

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

      1. *-commutative 48.7%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 96.2%

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

   6. Taylor expanded in c around inf 96.2%

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

   7. Taylor expanded in c around 0 94.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.1)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (*
    c
    (+
     (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
     (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.1) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.1], 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[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / 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 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.10000000000000001

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

   3. Simplified 80.8%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.7%

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

      1. *-commutative 48.7%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in c around 0 93.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.1)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ c (- b)) (* a (/ (pow c 2.0) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.1) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c / -b) - (a * (pow(c, 2.0) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / (-b)), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.10000000000000001

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

   3. Simplified 80.8%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.7%

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

      1. *-commutative 48.7%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 89.0%

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

      1. mul-1-neg 89.0%

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

      2. unsub-neg 89.0%

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

      3. mul-1-neg 89.0%

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

      4. distribute-neg-frac2 89.0%

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

      5. associate-/l* 89.0%

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

   7. Simplified 89.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.1) t_0 (- (/ c (- b)) (* a (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = (c / -b) - (a * (pow(c, 2.0) / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.1d0)) then
        tmp = t_0
    else
        tmp = (c / -b) - (a * ((c ** 2.0d0) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = (c / -b) - (a * (Math.pow(c, 2.0) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.1:
		tmp = t_0
	else:
		tmp = (c / -b) - (a * (math.pow(c, 2.0) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.1)
		tmp = t_0;
	else
		tmp = (c / -b) - (a * ((c ^ 2.0) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], t$95$0, N[(N[(c / (-b)), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.10000000000000001

   1. Initial program 80.7%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.7%

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

      1. *-commutative 48.7%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 89.0%

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

      1. mul-1-neg 89.0%

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

      2. unsub-neg 89.0%

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

      3. mul-1-neg 89.0%

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

      4. distribute-neg-frac2 89.0%

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

      5. associate-/l* 89.0%

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

   7. Simplified 89.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.1) t_0 (/ (fma a (pow (/ c (- b)) 2.0) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = fma(a, pow((c / -b), 2.0), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = t_0;
	else
		tmp = Float64(fma(a, (Float64(c / Float64(-b)) ^ 2.0), c) / Float64(-b));
	end
	return 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, -0.1], t$95$0, N[(N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.10000000000000001

   1. Initial program 80.7%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.7%

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

      1. *-commutative 48.7%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 89.0%

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

      1. mul-1-neg 89.0%

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

      2. unsub-neg 89.0%

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

      3. mul-1-neg 89.0%

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

      4. distribute-neg-frac2 89.0%

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

      5. associate-/l* 89.0%

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

   7. Simplified 89.0%

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

   8. Taylor expanded in b around inf 89.0%

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

      1. distribute-lft-out 89.0%

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

      2. associate-*r/ 89.0%

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

      3. mul-1-neg 89.0%

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

      4. distribute-neg-frac2 89.0%

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

      5. +-commutative 89.0%

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

      6. associate-/l* 89.0%

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

      7. fma-define 89.0%

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

      8. unpow2 89.0%

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

      9. unpow2 89.0%

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

      10. times-frac 89.0%

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

      11. sqr-neg 89.0%

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

      12. distribute-frac-neg2 89.0%

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

      13. distribute-frac-neg2 89.0%

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

      14. unpow2 89.0%

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

   10. Simplified 89.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-2 \cdot \left(\frac{a}{b} + \left(a \cdot a\right) \cdot \frac{c}{{b}^{3}}\right)\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.1)
     t_0
     (/ (* c (* -2.0 (+ (/ a b) (* (* a a) (/ c (pow b 3.0)))))) (* a 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 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = (c * (-2.0 * ((a / b) + ((a * a) * (c / pow(b, 3.0)))))) / (a * 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 <= (-0.1d0)) then
        tmp = t_0
    else
        tmp = (c * ((-2.0d0) * ((a / b) + ((a * a) * (c / (b ** 3.0d0)))))) / (a * 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 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = (c * (-2.0 * ((a / b) + ((a * a) * (c / Math.pow(b, 3.0)))))) / (a * 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 <= -0.1:
		tmp = t_0
	else:
		tmp = (c * (-2.0 * ((a / b) + ((a * a) * (c / math.pow(b, 3.0)))))) / (a * 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 <= -0.1)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * Float64(-2.0 * Float64(Float64(a / b) + Float64(Float64(a * a) * Float64(c / (b ^ 3.0)))))) / Float64(a * 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 <= -0.1)
		tmp = t_0;
	else
		tmp = (c * (-2.0 * ((a / b) + ((a * a) * (c / (b ^ 3.0)))))) / (a * 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, -0.1], t$95$0, N[(N[(c * N[(-2.0 * N[(N[(a / b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.10000000000000001

   1. Initial program 80.7%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.7%

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

      1. *-commutative 48.7%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in c around 0 88.9%

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

      1. distribute-lft-out 88.9%

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

      2. associate-/l* 88.9%

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

   7. Simplified 88.9%

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

      1. unpow2 88.9%

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

   9. Applied egg-rr 88.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-2 \cdot \left(\frac{a}{b} + \left(a \cdot a\right) \cdot \frac{c}{{b}^{3}}\right)\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.1) t_0 (* c (- (/ -1.0 b) (/ (* a c) (pow b 3.0)))))))

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 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = c * ((-1.0 / b) - ((a * c) / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.1d0)) then
        tmp = t_0
    else
        tmp = c * (((-1.0d0) / b) - ((a * c) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = c * ((-1.0 / b) - ((a * c) / Math.pow(b, 3.0)));
	}
	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 <= -0.1:
		tmp = t_0
	else:
		tmp = c * ((-1.0 / b) - ((a * c) / math.pow(b, 3.0)))
	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 <= -0.1)
		tmp = t_0;
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(a * c) / (b ^ 3.0))));
	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 <= -0.1)
		tmp = t_0;
	else
		tmp = c * ((-1.0 / b) - ((a * c) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], t$95$0, N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.10000000000000001

   1. Initial program 80.7%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.7%

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

      1. *-commutative 48.7%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in c around 0 88.8%

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

      1. associate-*r/ 88.8%

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

      2. neg-mul-1 88.8%

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

      3. distribute-rgt-neg-in 88.8%

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

   7. Simplified 88.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (/ (* a c) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((a * c) / pow(b, 3.0)));
}

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 * (((-1.0d0) / b) - ((a * c) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((a * c) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - ((a * c) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(a * c) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - ((a * c) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

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

   1. *-commutative 55.5%

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

3. Simplified 55.5%

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

5. Taylor expanded in c around 0 82.7%

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

   1. associate-*r/ 82.7%

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

   2. neg-mul-1 82.7%

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

   3. distribute-rgt-neg-in 82.7%

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

7. Simplified 82.7%

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

8. Final simplification 82.7%

   \[\leadsto c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right) \]
9. Add Preprocessing

Alternative 13: 64.6% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

   1. *-commutative 55.5%

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

3. Simplified 55.5%

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

5. Taylor expanded in b around inf 64.8%

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

   1. associate-*r/ 64.8%

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

   2. mul-1-neg 64.8%

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

7. Simplified 64.8%

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

8. Final simplification 64.8%

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

Alternative 14: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 55.5%

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

   1. *-commutative 55.5%

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

3. Simplified 55.5%

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

   1. *-commutative 55.5%

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

   2. metadata-eval 55.5%

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

   3. distribute-lft-neg-in 55.5%

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

   4. distribute-rgt-neg-in 55.5%

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

   5. *-commutative 55.5%

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

   6. fmm-def 55.5%

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

   7. add-sqr-sqrt 55.5%

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

   8. difference-of-squares 55.5%

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

   9. associate-*l* 55.5%

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

   10. sqrt-prod 55.5%

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

   11. metadata-eval 55.5%

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

   12. associate-*l* 55.5%

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

   13. sqrt-prod 55.5%

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

   14. metadata-eval 55.5%

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

6. Applied egg-rr 55.5%

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

7. Taylor expanded in b around inf 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt-out 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-rgt 3.2%

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

   5. metadata-eval 3.2%

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

9. Simplified 3.2%

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

10. Taylor expanded in a around 0 3.2%

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

Reproduce

herbie shell --seed 2024181 
(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)))