Quadratic roots, narrow range

Percentage Accurate: 55.2% → 91.7%

Time: 16.4s

Alternatives: 11

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% 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: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), {b}^{-2}, 1\right)\\ \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({b}^{2}, t\_0, -{b}^{2}\right)}{\mathsf{fma}\left(\left|b\right|, \sqrt{t\_0}, b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a (* c -4.0)) (pow b -2.0) 1.0)))
   (if (<= b 6.0)
     (/
      (/ (fma (pow b 2.0) t_0 (- (pow b 2.0))) (fma (fabs b) (sqrt t_0) b))
      (* 2.0 a))
     (-
      (*
       a
       (-
        (*
         a
         (+
          (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
          (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))
        (/ (* c c) (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * (c * -4.0)), pow(b, -2.0), 1.0);
	double tmp;
	if (b <= 6.0) {
		tmp = (fma(pow(b, 2.0), t_0, -pow(b, 2.0)) / fma(fabs(b), sqrt(t_0), b)) / (2.0 * a);
	} else {
		tmp = (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))) - ((c * c) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * Float64(c * -4.0)), (b ^ -2.0), 1.0)
	tmp = 0.0
	if (b <= 6.0)
		tmp = Float64(Float64(fma((b ^ 2.0), t_0, Float64(-(b ^ 2.0))) / fma(abs(b), sqrt(t_0), b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))) - Float64(Float64(c * c) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] * N[Power[b, -2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[b, 6.0], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] * t$95$0 + (-N[Power[b, 2.0], $MachinePrecision])), $MachinePrecision] / N[(N[Abs[b], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in b around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. associate-*r* 85.4%

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

      3. *-commutative 85.4%

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

   7. Simplified 85.4%

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

      1. flip-- 85.2%

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

      2. add-sqr-sqrt 86.5%

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

      3. unpow2 86.5%

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

      4. fmm-def 87.4%

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

      5. +-commutative 87.4%

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

      6. div-inv 87.4%

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

      7. fma-define 87.4%

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

      8. associate-*l* 87.4%

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

      9. pow-flip 87.4%

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

      10. metadata-eval 87.4%

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

   9. Applied egg-rr 87.4%

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

       1. *-commutative 87.4%

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

       2. unpow2 87.4%

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

       3. rem-sqrt-square 87.4%

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

       4. *-commutative 87.4%

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

   11. Simplified 87.4%

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

   if 6 < b

   1. Initial program 49.9%

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

      1. *-commutative 49.9%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in c around 0 94.0%

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

      1. unpow2 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-*r/ 94.0%

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

      2. neg-mul-1 94.0%

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

   10. Applied egg-rr 94.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({b}^{2}, \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), {b}^{-2}, 1\right), -{b}^{2}\right)}{\mathsf{fma}\left(\left|b\right|, \sqrt{\mathsf{fma}\left(a \cdot \left(c \cdot -4\right), {b}^{-2}, 1\right)}, b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)\\ \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, t\_0, -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{a}, \sqrt{t\_0}, b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c -4.0 (/ (pow b 2.0) a))))
   (if (<= b 6.2)
     (/ (/ (fma a t_0 (- (pow b 2.0))) (fma (sqrt a) (sqrt t_0) b)) (* 2.0 a))
     (-
      (*
       a
       (-
        (*
         a
         (+
          (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
          (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))
        (/ (* c c) (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, -4.0, (pow(b, 2.0) / a));
	double tmp;
	if (b <= 6.2) {
		tmp = (fma(a, t_0, -pow(b, 2.0)) / fma(sqrt(a), sqrt(t_0), b)) / (2.0 * a);
	} else {
		tmp = (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))) - ((c * c) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, -4.0, Float64((b ^ 2.0) / a))
	tmp = 0.0
	if (b <= 6.2)
		tmp = Float64(Float64(fma(a, t_0, Float64(-(b ^ 2.0))) / fma(sqrt(a), sqrt(t_0), b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))) - Float64(Float64(c * c) / (b ^ 3.0)))) - Float64(c / b));
	end
	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[b, 6.2], 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[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000018

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in a around inf 85.1%

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

      1. flip-- 84.8%

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

      2. unpow2 84.8%

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

      3. add-sqr-sqrt 85.7%

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

      4. fmm-def 86.1%

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

      5. *-commutative 86.1%

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

      6. fma-define 86.1%

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

      7. sqrt-prod 86.1%

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

      8. fma-define 86.1%

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

      9. *-commutative 86.1%

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

      10. fma-define 86.1%

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

   7. Applied egg-rr 86.1%

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

   if 6.20000000000000018 < b

   1. Initial program 49.9%

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

      1. *-commutative 49.9%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in c around 0 94.0%

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

      1. unpow2 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-*r/ 94.0%

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

      2. neg-mul-1 94.0%

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

   10. Applied egg-rr 94.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right), -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{a}, \sqrt{\mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)}, b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)\\ \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\frac{a \cdot t\_0 - {b}^{2}}{\mathsf{fma}\left(\sqrt{a}, \sqrt{t\_0}, b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c -4.0 (/ (pow b 2.0) a))))
   (if (<= b 6.0)
     (/ (/ (- (* a t_0) (pow b 2.0)) (fma (sqrt a) (sqrt t_0) b)) (* 2.0 a))
     (-
      (*
       a
       (-
        (*
         a
         (+
          (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
          (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))
        (/ (* c c) (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, -4.0, (pow(b, 2.0) / a));
	double tmp;
	if (b <= 6.0) {
		tmp = (((a * t_0) - pow(b, 2.0)) / fma(sqrt(a), sqrt(t_0), b)) / (2.0 * a);
	} else {
		tmp = (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))) - ((c * c) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, -4.0, Float64((b ^ 2.0) / a))
	tmp = 0.0
	if (b <= 6.0)
		tmp = Float64(Float64(Float64(Float64(a * t_0) - (b ^ 2.0)) / fma(sqrt(a), sqrt(t_0), b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))) - Float64(Float64(c * c) / (b ^ 3.0)))) - Float64(c / b));
	end
	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[b, 6.0], N[(N[(N[(N[(a * t$95$0), $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[a], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in a around inf 85.1%

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

      1. flip-- 84.8%

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

      2. unpow2 84.8%

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

      3. add-sqr-sqrt 85.7%

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

      4. fmm-def 86.1%

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

      5. *-commutative 86.1%

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

      6. fma-define 86.1%

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

      7. sqrt-prod 86.1%

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

      8. fma-define 86.1%

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

      9. *-commutative 86.1%

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

      10. fma-define 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. fmm-undef 85.7%

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

   9. Simplified 85.7%

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

   if 6 < b

   1. Initial program 49.9%

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

      1. *-commutative 49.9%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in c around 0 94.0%

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

      1. unpow2 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-*r/ 94.0%

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

      2. neg-mul-1 94.0%

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

   10. Applied egg-rr 94.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\frac{a \cdot \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right) - {b}^{2}}{\mathsf{fma}\left(\sqrt{a}, \sqrt{\mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)}, b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (-
    (*
     a
     (-
      (*
       a
       (+
        (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
        (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))
      (/ (* c c) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))) - ((c * c) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))) - Float64(Float64(c * c) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

   3. Simplified 85.6%

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

   if 6 < b

   1. Initial program 49.9%

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

      1. *-commutative 49.9%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in c around 0 94.0%

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

      1. unpow2 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-*r/ 94.0%

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

      2. neg-mul-1 94.0%

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

   10. Applied egg-rr 94.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.0102:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(b, \sqrt{1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.0102)
   (* 0.5 (/ (fma b (sqrt (+ 1.0 (* -4.0 (/ (* a c) (pow b 2.0))))) (- b)) a))
   (/ (- (- c) (* a (pow (/ c b) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.0102) {
		tmp = 0.5 * (fma(b, sqrt((1.0 + (-4.0 * ((a * c) / pow(b, 2.0))))), -b) / a);
	} else {
		tmp = (-c - (a * pow((c / b), 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -0.0102)
		tmp = Float64(0.5 * Float64(fma(b, sqrt(Float64(1.0 + Float64(-4.0 * Float64(Float64(a * c) / (b ^ 2.0))))), Float64(-b)) / a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / b) ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.0102], N[(0.5 * N[(N[(b * N[Sqrt[N[(1.0 + N[(-4.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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)) < -0.010200000000000001

   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 81.0%

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

   5. Taylor expanded in b around inf 80.7%

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

      1. associate-*r/ 80.7%

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

      2. associate-*r* 80.7%

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

      3. *-commutative 80.7%

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

   7. Simplified 80.7%

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

      1. log1p-expm1-u 53.9%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{\left(a \cdot -4\right) \cdot c}{{b}^{2}}\right)} - b}{a \cdot 2}\right)\right)} \]

      2. log1p-undefine 53.8%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \left(1 + \frac{\left(a \cdot -4\right) \cdot c}{{b}^{2}}\right)} - b}{a \cdot 2}\right)\right)} \]

      3. +-commutative 53.8%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \color{blue}{\left(\frac{\left(a \cdot -4\right) \cdot c}{{b}^{2}} + 1\right)}} - b}{a \cdot 2}\right)\right) \]

      4. div-inv 53.8%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \left(\color{blue}{\left(\left(a \cdot -4\right) \cdot c\right) \cdot \frac{1}{{b}^{2}}} + 1\right)} - b}{a \cdot 2}\right)\right) \]

      5. fma-define 53.8%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \color{blue}{\mathsf{fma}\left(\left(a \cdot -4\right) \cdot c, \frac{1}{{b}^{2}}, 1\right)}} - b}{a \cdot 2}\right)\right) \]

      6. associate-*l* 53.8%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \mathsf{fma}\left(\color{blue}{a \cdot \left(-4 \cdot c\right)}, \frac{1}{{b}^{2}}, 1\right)} - b}{a \cdot 2}\right)\right) \]

      7. pow-flip 53.8%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \mathsf{fma}\left(a \cdot \left(-4 \cdot c\right), \color{blue}{{b}^{\left(-2\right)}}, 1\right)} - b}{a \cdot 2}\right)\right) \]

      8. metadata-eval 53.8%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \mathsf{fma}\left(a \cdot \left(-4 \cdot c\right), {b}^{\color{blue}{-2}}, 1\right)} - b}{a \cdot 2}\right)\right) \]

   9. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \mathsf{fma}\left(a \cdot \left(-4 \cdot c\right), {b}^{-2}, 1\right)} - b}{a \cdot 2}\right)\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 53.5%

          \[\leadsto \log \color{blue}{\left(\sqrt{1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \mathsf{fma}\left(a \cdot \left(-4 \cdot c\right), {b}^{-2}, 1\right)} - b}{a \cdot 2}\right)} \cdot \sqrt{1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \mathsf{fma}\left(a \cdot \left(-4 \cdot c\right), {b}^{-2}, 1\right)} - b}{a \cdot 2}\right)}\right)} \]

       2. log-prod 53.5%

          \[\leadsto \color{blue}{\log \left(\sqrt{1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \mathsf{fma}\left(a \cdot \left(-4 \cdot c\right), {b}^{-2}, 1\right)} - b}{a \cdot 2}\right)}\right) + \log \left(\sqrt{1 + \mathsf{expm1}\left(\frac{\sqrt{{b}^{2} \cdot \mathsf{fma}\left(a \cdot \left(-4 \cdot c\right), {b}^{-2}, 1\right)} - b}{a \cdot 2}\right)}\right)} \]

   11. Applied egg-rr 68.2%

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

       1. count-2 68.2%

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

   13. Simplified 68.2%

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

   14. Taylor expanded in c around inf 80.5%

       \[\leadsto \color{blue}{0.5 \cdot \frac{b \cdot \sqrt{1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}} - b}{a}} \]
   15. Step-by-step derivation

       1. fmm-def 81.2%

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

       2. *-commutative 81.2%

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

   16. Simplified 81.2%

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

   if -0.010200000000000001 < (/.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 46.1%

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

      1. *-commutative 46.1%

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

   3. Simplified 46.4%

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

   5. Taylor expanded in b around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. unsub-neg 88.5%

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

      3. mul-1-neg 88.5%

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

      4. associate-/l* 88.5%

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

   7. Simplified 88.5%

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

      1. expm1-log1p-u 70.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}}{b} \]

      2. div-inv 70.7%

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \color{blue}{\left({c}^{2} \cdot \frac{1}{{b}^{2}}\right)}\right)\right)}{b} \]

      3. pow-flip 70.7%

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot \color{blue}{{b}^{\left(-2\right)}}\right)\right)\right)}{b} \]

      4. metadata-eval 70.7%

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{\color{blue}{-2}}\right)\right)\right)}{b} \]

   9. Applied egg-rr 70.7%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)\right)}}{b} \]
   10. Step-by-step derivation

       1. expm1-undefine 59.6%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)} - 1}}{b} \]

       2. sub-neg 59.6%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)} + \left(-1\right)}}{b} \]

       3. log1p-undefine 59.6%

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

       4. rem-exp-log 77.4%

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

       5. sub-neg 77.4%

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

       6. distribute-neg-out 77.4%

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

       7. unsub-neg 77.4%

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

       8. metadata-eval 77.4%

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

   11. Simplified 77.4%

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

   12. Taylor expanded in a around 0 88.5%

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

       1. sub-neg 88.5%

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

       2. +-commutative 88.5%

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

       3. mul-1-neg 88.5%

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

       4. unsub-neg 88.5%

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

       5. associate-/l* 88.5%

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

       6. unpow2 88.5%

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

       7. unpow2 88.5%

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

       8. times-frac 88.5%

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

       9. unpow1 88.5%

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

       10. pow-plus 88.5%

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

       11. metadata-eval 88.5%

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

   14. Simplified 88.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.0102:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(b, \sqrt{1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (-
    (*
     a
     (-
      (/ (* -2.0 (* a (pow c 3.0))) (pow b 5.0))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (a * (((-2.0 * (a * pow(c, 3.0))) / pow(b, 5.0)) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64(Float64(Float64(-2.0 * Float64(a * (c ^ 3.0))) / (b ^ 5.0)) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(-2.0 * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $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 b < 6

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

   3. Simplified 85.6%

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

   if 6 < b

   1. Initial program 49.9%

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

      1. *-commutative 49.9%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in b around inf 49.8%

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

      1. associate-*r/ 49.8%

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

      2. associate-*r* 49.8%

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

      3. *-commutative 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in a around 0 91.9%

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

      1. neg-mul-1 91.9%

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

      2. +-commutative 91.9%

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

      3. unsub-neg 91.9%

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

      4. mul-1-neg 91.9%

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

      5. unsub-neg 91.9%

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

      6. associate-*r/ 91.9%

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

   10. Simplified 91.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.0102:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.0102)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (/ (- (- c) (* a (pow (/ c b) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.0102) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (-c - (a * pow((c / b), 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -0.0102)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / b) ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.0102], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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)) < -0.010200000000000001

   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 81.0%

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

   if -0.010200000000000001 < (/.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 46.1%

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

      1. *-commutative 46.1%

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

   3. Simplified 46.4%

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

   5. Taylor expanded in b around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. unsub-neg 88.5%

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

      3. mul-1-neg 88.5%

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

      4. associate-/l* 88.5%

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

   7. Simplified 88.5%

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

      1. expm1-log1p-u 70.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}}{b} \]

      2. div-inv 70.7%

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \color{blue}{\left({c}^{2} \cdot \frac{1}{{b}^{2}}\right)}\right)\right)}{b} \]

      3. pow-flip 70.7%

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot \color{blue}{{b}^{\left(-2\right)}}\right)\right)\right)}{b} \]

      4. metadata-eval 70.7%

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{\color{blue}{-2}}\right)\right)\right)}{b} \]

   9. Applied egg-rr 70.7%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)\right)}}{b} \]
   10. Step-by-step derivation

       1. expm1-undefine 59.6%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)} - 1}}{b} \]

       2. sub-neg 59.6%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)} + \left(-1\right)}}{b} \]

       3. log1p-undefine 59.6%

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

       4. rem-exp-log 77.4%

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

       5. sub-neg 77.4%

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

       6. distribute-neg-out 77.4%

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

       7. unsub-neg 77.4%

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

       8. metadata-eval 77.4%

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

   11. Simplified 77.4%

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

   12. Taylor expanded in a around 0 88.5%

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

       1. sub-neg 88.5%

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

       2. +-commutative 88.5%

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

       3. mul-1-neg 88.5%

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

       4. unsub-neg 88.5%

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

       5. associate-/l* 88.5%

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

       6. unpow2 88.5%

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

       7. unpow2 88.5%

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

       8. times-frac 88.5%

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

       9. unpow1 88.5%

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

       10. pow-plus 88.5%

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

       11. metadata-eval 88.5%

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

   14. Simplified 88.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.0102:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (*
    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 (b <= 6.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} 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 (b <= 6.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	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[b, 6.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $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 b < 6

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

   3. Simplified 85.6%

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

   if 6 < b

   1. Initial program 49.9%

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

      1. *-commutative 49.9%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in b around inf 91.7%

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

      1. fma-define 91.7%

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

      2. cube-prod 91.7%

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

      3. distribute-lft-out 91.7%

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

      4. *-commutative 91.7%

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

   7. Simplified 91.7%

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

   8. Taylor expanded in a around -inf 91.6%

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

      1. mul-1-neg 91.6%

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

      2. *-commutative 91.6%

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

      3. distribute-rgt-neg-in 91.6%

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

   10. Simplified 91.6%

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

   11. Taylor expanded in c around 0 91.8%

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

       1. fma-define 91.8%

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

       2. mul-1-neg 91.8%

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

       3. fmm-undef 91.8%

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

       4. *-commutative 91.8%

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

   13. Simplified 91.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.0)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
   (/ (- (- c) (* a (pow (/ c b) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (-c - (a * pow((c / b), 2.0))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    else
        tmp = (-c - (a * ((c / b) ** 2.0d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (-c - (a * Math.pow((c / b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	else:
		tmp = (-c - (a * math.pow((c / b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / b) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	else
		tmp = (-c - (a * ((c / b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6

   1. Initial program 85.4%

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

   if 6 < b

   1. Initial program 49.9%

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

      1. *-commutative 49.9%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in b around inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. unsub-neg 86.3%

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

      3. mul-1-neg 86.3%

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

      4. associate-/l* 86.3%

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

   7. Simplified 86.3%

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

      1. expm1-log1p-u 53.6%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}}{b} \]

      2. div-inv 53.6%

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \color{blue}{\left({c}^{2} \cdot \frac{1}{{b}^{2}}\right)}\right)\right)}{b} \]

      3. pow-flip 53.6%

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot \color{blue}{{b}^{\left(-2\right)}}\right)\right)\right)}{b} \]

      4. metadata-eval 53.6%

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{\color{blue}{-2}}\right)\right)\right)}{b} \]

   9. Applied egg-rr 53.6%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)\right)}}{b} \]
   10. Step-by-step derivation

       1. expm1-undefine 44.8%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)} - 1}}{b} \]

       2. sub-neg 44.8%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)} + \left(-1\right)}}{b} \]

       3. log1p-undefine 44.8%

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

       4. rem-exp-log 77.5%

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

       5. sub-neg 77.5%

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

       6. distribute-neg-out 77.5%

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

       7. unsub-neg 77.5%

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

       8. metadata-eval 77.5%

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

   11. Simplified 77.5%

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

   12. Taylor expanded in a around 0 86.3%

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

       1. sub-neg 86.3%

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

       2. +-commutative 86.3%

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

       3. mul-1-neg 86.3%

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

       4. unsub-neg 86.3%

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

       5. associate-/l* 86.3%

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

       6. unpow2 86.3%

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

       7. unpow2 86.3%

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

       8. times-frac 86.3%

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

       9. unpow1 86.3%

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

       10. pow-plus 86.3%

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

       11. metadata-eval 86.3%

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

   14. Simplified 86.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. *-commutative 56.5%

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

3. Simplified 56.8%

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

5. Taylor expanded in b around inf 80.0%

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

   1. mul-1-neg 80.0%

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

   2. unsub-neg 80.0%

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

   3. mul-1-neg 80.0%

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

   4. associate-/l* 80.0%

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

7. Simplified 80.0%

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

   1. expm1-log1p-u 52.1%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}}{b} \]

   2. div-inv 52.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \color{blue}{\left({c}^{2} \cdot \frac{1}{{b}^{2}}\right)}\right)\right)}{b} \]

   3. pow-flip 52.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot \color{blue}{{b}^{\left(-2\right)}}\right)\right)\right)}{b} \]

   4. metadata-eval 52.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{\color{blue}{-2}}\right)\right)\right)}{b} \]

9. Applied egg-rr 52.1%

   \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)\right)}}{b} \]
10. Step-by-step derivation

    1. expm1-undefine 44.3%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)} - 1}}{b} \]

    2. sub-neg 44.3%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-c\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right)} + \left(-1\right)}}{b} \]

    3. log1p-undefine 44.3%

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

    4. rem-exp-log 72.2%

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

    5. sub-neg 72.2%

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

    6. distribute-neg-out 72.2%

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

    7. unsub-neg 72.2%

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

    8. metadata-eval 72.2%

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

11. Simplified 72.2%

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

12. Taylor expanded in a around 0 80.0%

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

    1. sub-neg 80.0%

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

    2. +-commutative 80.0%

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

    3. mul-1-neg 80.0%

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

    4. unsub-neg 80.0%

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

    5. associate-/l* 80.0%

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

    6. unpow2 80.0%

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

    7. unpow2 80.0%

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

    8. times-frac 80.0%

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

    9. unpow1 80.0%

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

    10. pow-plus 80.0%

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

    11. metadata-eval 80.0%

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

14. Simplified 80.0%

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

Alternative 11: 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 56.5%

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

   1. *-commutative 56.5%

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

3. Simplified 56.8%

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

5. Taylor expanded in b around inf 63.4%

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

   1. associate-*r/ 63.4%

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

   2. mul-1-neg 63.4%

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

7. Simplified 63.4%

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

8. Final simplification 63.4%

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

Reproduce

herbie shell --seed 2024172 
(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)))