Quadratic roots, narrow range

Percentage Accurate: 55.2% → 92.0%

Time: 17.2s

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: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, t\_0, -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{a}, \sqrt{t\_0}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right) - c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c -4.0 (/ (pow b 2.0) a))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -30.0)
     (/ (/ (fma a t_0 (- (pow b 2.0))) (fma (sqrt a) (sqrt t_0) b)) (* a 2.0))
     (/
      (fma
       -2.0
       (/ (* (pow c 3.0) (pow a 2.0)) (pow b 4.0))
       (-
        (-
         (* -0.25 (/ (* (pow (* a c) 4.0) (* 20.0 (pow b -6.0))) a))
         (* a (* (/ c b) (/ c b))))
        c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, -4.0, (pow(b, 2.0) / a));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = (fma(a, t_0, -pow(b, 2.0)) / fma(sqrt(a), sqrt(t_0), b)) / (a * 2.0);
	} else {
		tmp = fma(-2.0, ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 4.0)), (((-0.25 * ((pow((a * c), 4.0) * (20.0 * pow(b, -6.0))) / a)) - (a * ((c / b) * (c / b)))) - c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, -4.0, Float64((b ^ 2.0) / a))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -30.0)
		tmp = Float64(Float64(fma(a, t_0, Float64(-(b ^ 2.0))) / fma(sqrt(a), sqrt(t_0), b)) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-2.0, Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 4.0)), Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) * Float64(20.0 * (b ^ -6.0))) / a)) - Float64(a * Float64(Float64(c / b) * Float64(c / b)))) - c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * -4.0 + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -30.0], N[(N[(N[(a * t$95$0 + (-N[Power[b, 2.0], $MachinePrecision])), $MachinePrecision] / N[(N[Sqrt[a], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $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)) < -30

   1. Initial program 86.9%

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

      1. +-commutative 86.9%

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

      2. sqr-neg 86.9%

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

      3. unsub-neg 86.9%

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

      4. sqr-neg 86.9%

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

      5. sub-neg 86.9%

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

      6. +-commutative 86.9%

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

      7. *-commutative 86.9%

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

      8. associate-*r* 86.9%

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

      9. distribute-rgt-neg-in 86.9%

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

      10. fma-define 87.0%

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

      11. *-commutative 87.0%

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

      12. distribute-rgt-neg-in 87.0%

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

      13. metadata-eval 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in a around inf 86.4%

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

      1. flip-- 86.9%

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

      2. unpow2 86.9%

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

      3. add-sqr-sqrt 87.3%

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

      4. fma-neg 88.2%

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

      5. *-commutative 88.2%

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

      6. fma-define 88.2%

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

      7. sqrt-prod 88.2%

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

      8. fma-define 88.3%

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

      9. *-commutative 88.3%

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

      10. fma-define 88.3%

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

   7. Applied egg-rr 88.3%

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

   if -30 < (/.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 51.1%

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

      1. *-commutative 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in b around inf 94.3%

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

   7. Simplified 94.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]
   8. Step-by-step derivation

      1. associate-*l/ 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \color{blue}{\frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{20}{{b}^{6}}}{a}} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      2. pow-prod-down 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \frac{20}{{b}^{6}}}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      3. div-inv 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \color{blue}{\left(20 \cdot \frac{1}{{b}^{6}}\right)}}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      4. pow-flip 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot \color{blue}{{b}^{\left(-6\right)}}\right)}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      5. metadata-eval 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{\color{blue}{-6}}\right)}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

   9. Applied egg-rr 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a}} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]
   10. Step-by-step derivation

       1. associate-/l* 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}\right) - c\right)}{b} \]

   11. Applied egg-rr 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}\right) - c\right)}{b} \]
   12. Step-by-step derivation

       1. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right) - c\right)}{b} \]

       2. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right) - c\right)}{b} \]

       3. times-frac 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right) - c\right)}{b} \]

       4. sqr-neg 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\left(-\frac{c}{b}\right) \cdot \left(-\frac{c}{b}\right)\right)}\right) - c\right)}{b} \]

       5. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{{\left(-\frac{c}{b}\right)}^{2}}\right) - c\right)}{b} \]

       6. distribute-neg-frac2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot {\color{blue}{\left(\frac{c}{-b}\right)}}^{2}\right) - c\right)}{b} \]

   13. Simplified 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - \color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2}}\right) - c\right)}{b} \]
   14. Step-by-step derivation

       1. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\frac{c}{-b} \cdot \frac{c}{-b}\right)}\right) - c\right)}{b} \]

   15. Applied egg-rr 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\frac{c}{-b} \cdot \frac{c}{-b}\right)}\right) - c\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right), -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{a}, \sqrt{\mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right) - c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{\mathsf{fma}\left(\sqrt{a}, \sqrt{\mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right) - c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -30.0)
   (/
    (/
     (- (fma a (* c -4.0) (pow b 2.0)) (pow b 2.0))
     (fma (sqrt a) (sqrt (fma c -4.0 (/ (pow b 2.0) a))) b))
    (* a 2.0))
   (/
    (fma
     -2.0
     (/ (* (pow c 3.0) (pow a 2.0)) (pow b 4.0))
     (-
      (-
       (* -0.25 (/ (* (pow (* a c) 4.0) (* 20.0 (pow b -6.0))) a))
       (* a (* (/ c b) (/ c b))))
      c))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = ((fma(a, (c * -4.0), pow(b, 2.0)) - pow(b, 2.0)) / fma(sqrt(a), sqrt(fma(c, -4.0, (pow(b, 2.0) / a))), b)) / (a * 2.0);
	} else {
		tmp = fma(-2.0, ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 4.0)), (((-0.25 * ((pow((a * c), 4.0) * (20.0 * pow(b, -6.0))) / a)) - (a * ((c / b) * (c / b)))) - c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -30.0)
		tmp = Float64(Float64(Float64(fma(a, Float64(c * -4.0), (b ^ 2.0)) - (b ^ 2.0)) / fma(sqrt(a), sqrt(fma(c, -4.0, Float64((b ^ 2.0) / a))), b)) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-2.0, Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 4.0)), Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) * Float64(20.0 * (b ^ -6.0))) / a)) - Float64(a * Float64(Float64(c / b) * Float64(c / b)))) - c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -30.0], N[(N[(N[(N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[a], $MachinePrecision] * N[Sqrt[N[(c * -4.0 + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $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)) < -30

   1. Initial program 86.9%

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

      1. +-commutative 86.9%

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

      2. sqr-neg 86.9%

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

      3. unsub-neg 86.9%

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

      4. sqr-neg 86.9%

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

      5. sub-neg 86.9%

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

      6. +-commutative 86.9%

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

      7. *-commutative 86.9%

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

      8. associate-*r* 86.9%

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

      9. distribute-rgt-neg-in 86.9%

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

      10. fma-define 87.0%

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

      11. *-commutative 87.0%

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

      12. distribute-rgt-neg-in 87.0%

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

      13. metadata-eval 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in a around inf 86.4%

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

      1. flip-- 86.9%

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

      2. unpow2 86.9%

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

      3. add-sqr-sqrt 87.3%

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

      4. fma-neg 88.2%

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

      5. *-commutative 88.2%

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

      6. fma-define 88.2%

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

      7. sqrt-prod 88.2%

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

      8. fma-define 88.3%

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

      9. *-commutative 88.3%

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

      10. fma-define 88.3%

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

   7. Applied egg-rr 88.3%

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

      1. fma-undefine 87.3%

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

      2. unsub-neg 87.3%

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

      3. fma-undefine 87.3%

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

      4. distribute-lft-out 87.8%

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

      5. fma-define 87.9%

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

      6. *-commutative 87.9%

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

      7. associate-*l/ 87.8%

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

      8. associate-/l* 88.1%

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

      9. *-inverses 88.1%

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

      10. *-rgt-identity 88.1%

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

   9. Simplified 88.1%

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

   if -30 < (/.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 51.1%

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

      1. *-commutative 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in b around inf 94.3%

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

   7. Simplified 94.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]
   8. Step-by-step derivation

      1. associate-*l/ 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \color{blue}{\frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{20}{{b}^{6}}}{a}} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      2. pow-prod-down 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \frac{20}{{b}^{6}}}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      3. div-inv 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \color{blue}{\left(20 \cdot \frac{1}{{b}^{6}}\right)}}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      4. pow-flip 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot \color{blue}{{b}^{\left(-6\right)}}\right)}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      5. metadata-eval 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{\color{blue}{-6}}\right)}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

   9. Applied egg-rr 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a}} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]
   10. Step-by-step derivation

       1. associate-/l* 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}\right) - c\right)}{b} \]

   11. Applied egg-rr 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}\right) - c\right)}{b} \]
   12. Step-by-step derivation

       1. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right) - c\right)}{b} \]

       2. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right) - c\right)}{b} \]

       3. times-frac 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right) - c\right)}{b} \]

       4. sqr-neg 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\left(-\frac{c}{b}\right) \cdot \left(-\frac{c}{b}\right)\right)}\right) - c\right)}{b} \]

       5. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{{\left(-\frac{c}{b}\right)}^{2}}\right) - c\right)}{b} \]

       6. distribute-neg-frac2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot {\color{blue}{\left(\frac{c}{-b}\right)}}^{2}\right) - c\right)}{b} \]

   13. Simplified 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - \color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2}}\right) - c\right)}{b} \]
   14. Step-by-step derivation

       1. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\frac{c}{-b} \cdot \frac{c}{-b}\right)}\right) - c\right)}{b} \]

   15. Applied egg-rr 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\frac{c}{-b} \cdot \frac{c}{-b}\right)}\right) - c\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{\mathsf{fma}\left(\sqrt{a}, \sqrt{\mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right) - c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right) - c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -30.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (fma
     -2.0
     (/ (* (pow c 3.0) (pow a 2.0)) (pow b 4.0))
     (-
      (-
       (* -0.25 (/ (* (pow (* a c) 4.0) (* 20.0 (pow b -6.0))) a))
       (* a (* (/ c b) (/ c b))))
      c))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(-2.0, ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 4.0)), (((-0.25 * ((pow((a * c), 4.0) * (20.0 * pow(b, -6.0))) / a)) - (a * ((c / b) * (c / b)))) - c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -30.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-2.0, Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 4.0)), Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) * Float64(20.0 * (b ^ -6.0))) / a)) - Float64(a * Float64(Float64(c / b) * Float64(c / b)))) - c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -30.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $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)) < -30

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      2. +-commutative 86.9%

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

      3. sqr-neg 86.9%

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

      4. unsub-neg 86.9%

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

      5. sqr-neg 86.9%

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

      6. fma-neg 87.1%

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

      7. distribute-lft-neg-in 87.1%

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

      8. *-commutative 87.1%

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

      9. *-commutative 87.1%

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

      10. distribute-rgt-neg-in 87.1%

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

      11. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

   if -30 < (/.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 51.1%

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

      1. *-commutative 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in b around inf 94.3%

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

   7. Simplified 94.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]
   8. Step-by-step derivation

      1. associate-*l/ 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \color{blue}{\frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{20}{{b}^{6}}}{a}} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      2. pow-prod-down 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \frac{20}{{b}^{6}}}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      3. div-inv 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \color{blue}{\left(20 \cdot \frac{1}{{b}^{6}}\right)}}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      4. pow-flip 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot \color{blue}{{b}^{\left(-6\right)}}\right)}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

      5. metadata-eval 94.3%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{\color{blue}{-6}}\right)}{a} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]

   9. Applied egg-rr 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a}} - \frac{a \cdot {c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]
   10. Step-by-step derivation

       1. associate-/l* 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}\right) - c\right)}{b} \]

   11. Applied egg-rr 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}\right) - c\right)}{b} \]
   12. Step-by-step derivation

       1. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right) - c\right)}{b} \]

       2. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right) - c\right)}{b} \]

       3. times-frac 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right) - c\right)}{b} \]

       4. sqr-neg 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\left(-\frac{c}{b}\right) \cdot \left(-\frac{c}{b}\right)\right)}\right) - c\right)}{b} \]

       5. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{{\left(-\frac{c}{b}\right)}^{2}}\right) - c\right)}{b} \]

       6. distribute-neg-frac2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot {\color{blue}{\left(\frac{c}{-b}\right)}}^{2}\right) - c\right)}{b} \]

   13. Simplified 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - \color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2}}\right) - c\right)}{b} \]
   14. Step-by-step derivation

       1. unpow2 94.3%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\frac{c}{-b} \cdot \frac{c}{-b}\right)}\right) - c\right)}{b} \]

   15. Applied egg-rr 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \color{blue}{\left(\frac{c}{-b} \cdot \frac{c}{-b}\right)}\right) - c\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \left(20 \cdot {b}^{-6}\right)}{a} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right) - c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -30.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (*
     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 (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} 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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -30.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(-2.0 * Float64(a * Float64((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[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -30.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-2.0 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -30

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      2. +-commutative 86.9%

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

      3. sqr-neg 86.9%

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

      4. unsub-neg 86.9%

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

      5. sqr-neg 86.9%

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

      6. fma-neg 87.1%

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

      7. distribute-lft-neg-in 87.1%

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

      8. *-commutative 87.1%

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

      9. *-commutative 87.1%

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

      10. distribute-rgt-neg-in 87.1%

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

      11. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

   if -30 < (/.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 51.1%

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

      1. *-commutative 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in a around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. mul-1-neg 91.9%

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

      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-/l* 91.9%

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

   7. Simplified 91.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -30.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (*
    c
    (+
     (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
     (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -30.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -30.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -30

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      2. +-commutative 86.9%

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

      3. sqr-neg 86.9%

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

      4. unsub-neg 86.9%

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

      5. sqr-neg 86.9%

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

      6. fma-neg 87.1%

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

      7. distribute-lft-neg-in 87.1%

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

      8. *-commutative 87.1%

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

      9. *-commutative 87.1%

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

      10. distribute-rgt-neg-in 87.1%

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

      11. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

   if -30 < (/.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 51.1%

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

      1. *-commutative 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in c around 0 91.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.014:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.014)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (* a (/ (* c (- c)) (pow b 3.0))) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.014) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (a * ((c * -c) / pow(b, 3.0))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.014)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(c * Float64(-c)) / (b ^ 3.0))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.014], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * (-c)), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0140000000000000003

   1. Initial program 79.0%

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

      1. *-commutative 79.0%

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

      2. +-commutative 79.0%

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

      3. sqr-neg 79.0%

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

      4. unsub-neg 79.0%

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

      5. sqr-neg 79.0%

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

      6. fma-neg 79.2%

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

      7. distribute-lft-neg-in 79.2%

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

      8. *-commutative 79.2%

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

      9. *-commutative 79.2%

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

      10. distribute-rgt-neg-in 79.2%

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

      11. metadata-eval 79.2%

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

   3. Simplified 79.2%

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

   if -0.0140000000000000003 < (/.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 45.2%

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

      1. *-commutative 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in a around 0 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. mul-1-neg 90.6%

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

      4. distribute-neg-frac2 90.6%

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

      5. associate-/l* 90.6%

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

   7. Simplified 90.6%

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

      1. unpow2 90.6%

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

   9. Applied egg-rr 90.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.014:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.014:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.014)
   (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
   (- (* a (/ (* c (- c)) (pow b 3.0))) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.014) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = (a * ((c * -c) / pow(b, 3.0))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.014)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(c * Float64(-c)) / (b ^ 3.0))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.014], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * (-c)), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0140000000000000003

   1. Initial program 79.0%

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

      1. +-commutative 79.0%

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

      2. sqr-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. sqr-neg 79.0%

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

      5. sub-neg 79.0%

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

      6. +-commutative 79.0%

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

      7. *-commutative 79.0%

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

      8. associate-*r* 79.0%

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

      9. distribute-rgt-neg-in 79.0%

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

      10. fma-define 79.0%

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

      11. *-commutative 79.0%

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

      12. distribute-rgt-neg-in 79.0%

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

      13. metadata-eval 79.0%

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

   3. Simplified 79.0%

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

   if -0.0140000000000000003 < (/.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 45.2%

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

      1. *-commutative 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in a around 0 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. mul-1-neg 90.6%

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

      4. distribute-neg-frac2 90.6%

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

      5. associate-/l* 90.6%

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

   7. Simplified 90.6%

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

      1. unpow2 90.6%

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

   9. Applied egg-rr 90.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.014:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.014:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.014) t_0 (- (* a (/ (* c (- c)) (pow b 3.0))) (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.014) {
		tmp = t_0;
	} else {
		tmp = (a * ((c * -c) / pow(b, 3.0))) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.014d0)) then
        tmp = t_0
    else
        tmp = (a * ((c * -c) / (b ** 3.0d0))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.014) {
		tmp = t_0;
	} else {
		tmp = (a * ((c * -c) / Math.pow(b, 3.0))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.014:
		tmp = t_0
	else:
		tmp = (a * ((c * -c) / math.pow(b, 3.0))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.014)
		tmp = t_0;
	else
		tmp = Float64(Float64(a * Float64(Float64(c * Float64(-c)) / (b ^ 3.0))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.014)
		tmp = t_0;
	else
		tmp = (a * ((c * -c) / (b ^ 3.0))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.014], t$95$0, N[(N[(a * N[(N[(c * (-c)), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0140000000000000003

   1. Initial program 79.0%

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

   if -0.0140000000000000003 < (/.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 45.2%

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

      1. *-commutative 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in a around 0 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. mul-1-neg 90.6%

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

      4. distribute-neg-frac2 90.6%

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

      5. associate-/l* 90.6%

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

   7. Simplified 90.6%

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

      1. unpow2 90.6%

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

   9. Applied egg-rr 90.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.014:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (a * ((c * -c) / pow(b, 3.0))) - (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 = (a * ((c * -c) / (b ** 3.0d0))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. *-commutative 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in a around 0 82.8%

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

   1. mul-1-neg 82.8%

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

   2. unsub-neg 82.8%

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

   3. mul-1-neg 82.8%

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

   4. distribute-neg-frac2 82.8%

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

   5. associate-/l* 82.8%

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

7. Simplified 82.8%

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

   1. unpow2 82.8%

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

9. Applied egg-rr 82.8%

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

10. Final simplification 82.8%

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

Alternative 10: 81.8% 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 / Float64(-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 54.6%

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

   1. +-commutative 54.6%

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

   2. sqr-neg 54.6%

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

   3. unsub-neg 54.6%

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

   4. sqr-neg 54.6%

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

   5. sub-neg 54.6%

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

   6. +-commutative 54.6%

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

   7. *-commutative 54.6%

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

   8. associate-*r* 54.6%

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

   9. distribute-rgt-neg-in 54.6%

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

   10. fma-define 54.6%

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

   11. *-commutative 54.6%

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

   12. distribute-rgt-neg-in 54.6%

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

   13. metadata-eval 54.6%

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

3. Simplified 54.6%

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

   1. div-sub 53.6%

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

   2. *-un-lft-identity 53.6%

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

   3. *-commutative 53.6%

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

   4. times-frac 53.6%

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

   5. metadata-eval 53.6%

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

   6. pow2 53.6%

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

   7. *-un-lft-identity 53.6%

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

   8. *-commutative 53.6%

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

   9. times-frac 53.6%

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

   10. metadata-eval 53.6%

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

6. Applied egg-rr 53.6%

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

   1. fma-undefine 53.6%

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

8. Applied egg-rr 53.6%

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

9. Taylor expanded in b around inf 82.8%

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

    1. mul-1-neg 82.8%

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

    2. unsub-neg 82.8%

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

    3. mul-1-neg 82.8%

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

    4. associate-/l* 82.8%

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

    5. unpow2 82.8%

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

    6. unpow2 82.8%

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

    7. times-frac 82.8%

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

    8. sqr-neg 82.8%

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

    9. distribute-frac-neg2 82.8%

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

    10. distribute-frac-neg2 82.8%

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

    11. unpow2 82.8%

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

11. Simplified 82.8%

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

Alternative 11: 64.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(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 54.6%

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

   1. *-commutative 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in b around inf 64.7%

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

   1. associate-*r/ 64.7%

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

   2. mul-1-neg 64.7%

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

7. Simplified 64.7%

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

8. Final simplification 64.7%

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

Reproduce

herbie shell --seed 2024108 
(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)))