Quadratic roots, narrow range

Percentage Accurate: 55.7% → 99.3%

Time: 15.9s

Alternatives: 8

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: 55.7% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-a}{\frac{\left(-b\right) - \sqrt{\frac{\mathsf{fma}\left({\left(a \cdot c\right)}^{3}, -64, {b}^{6}\right)}{\mathsf{fma}\left(a \cdot -4, c \cdot \left(c \cdot \left(a \cdot -4\right) - {b}^{2}\right), {b}^{4}\right)}}}{c \cdot -4}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (- a)
   (/
    (-
     (- b)
     (sqrt
      (/
       (fma (pow (* a c) 3.0) -64.0 (pow b 6.0))
       (fma (* a -4.0) (* c (- (* c (* a -4.0)) (pow b 2.0))) (pow b 4.0)))))
    (* c -4.0)))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (-a / ((-b - sqrt((fma(pow((a * c), 3.0), -64.0, pow(b, 6.0)) / fma((a * -4.0), (c * ((c * (a * -4.0)) - pow(b, 2.0))), pow(b, 4.0))))) / (c * -4.0))) / (a * 2.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-a) / Float64(Float64(Float64(-b) - sqrt(Float64(fma((Float64(a * c) ^ 3.0), -64.0, (b ^ 6.0)) / fma(Float64(a * -4.0), Float64(c * Float64(Float64(c * Float64(a * -4.0)) - (b ^ 2.0))), (b ^ 4.0))))) / Float64(c * -4.0))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := N[(N[((-a) / N[(N[((-b) - N[Sqrt[N[(N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] * -64.0 + N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] / N[(N[(a * -4.0), $MachinePrecision] * N[(c * N[(N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

3. Simplified 54.5%

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

   1. add-cube-cbrt 52.4%

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

   2. fma-neg 52.6%

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

   3. *-commutative 52.6%

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

   4. distribute-rgt-neg-in 52.6%

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

   5. distribute-lft-neg-in 52.6%

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

   6. metadata-eval 52.6%

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

   7. *-commutative 52.6%

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

   8. cbrt-unprod 53.7%

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

   9. pow2 53.7%

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

   10. pow2 53.7%

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

   11. pow-prod-up 53.4%

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

   12. metadata-eval 53.4%

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

   13. cbrt-prod 53.1%

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

   14. pow2 53.1%

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

   15. *-commutative 53.1%

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

   16. associate-*r* 53.1%

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

6. Applied egg-rr 53.1%

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

   1. flip-+ 53.0%

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

8. Applied egg-rr 56.2%

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

   1. associate--r+ 99.2%

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

10. Simplified 99.2%

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

    1. flip3-+ 99.2%

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

    2. pow-pow 99.2%

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

    3. metadata-eval 99.2%

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

    4. pow-prod-up 99.2%

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

    5. metadata-eval 99.2%

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

    6. pow2 99.2%

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

12. Applied egg-rr 99.2%

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

    1. associate-*r* 99.2%

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

    2. cube-prod 99.2%

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

    3. metadata-eval 99.2%

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

    4. unpow2 99.2%

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

    5. distribute-rgt-out-- 99.2%

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

    6. associate-*r* 99.2%

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

    7. *-commutative 99.2%

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

    8. associate-*r* 99.2%

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

    9. associate-*r* 99.2%

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

    10. *-commutative 99.2%

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

    11. associate-*r* 99.2%

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

14. Simplified 99.2%

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

16. Applied egg-rr 99.2%

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

    1. div-sub 99.2%

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

    2. +-inverses 99.2%

       \[\leadsto \frac{\color{blue}{0} - \frac{a \cdot \left(c \cdot -4\right)}{\left(-b\right) - \sqrt{\frac{\mathsf{fma}\left({\left(a \cdot c\right)}^{3}, -64, {b}^{6}\right)}{\mathsf{fma}\left(a \cdot -4, c \cdot \left(a \cdot \left(c \cdot -4\right) - {b}^{2}\right), {b}^{4}\right)}}}}{a \cdot 2} \]

    3. neg-sub0 99.2%

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

    4. associate-/l* 99.3%

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

18. Simplified 99.3%

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

19. Final simplification 99.3%

    \[\leadsto \frac{\frac{-a}{\frac{\left(-b\right) - \sqrt{\frac{\mathsf{fma}\left({\left(a \cdot c\right)}^{3}, -64, {b}^{6}\right)}{\mathsf{fma}\left(a \cdot -4, c \cdot \left(c \cdot \left(a \cdot -4\right) - {b}^{2}\right), {b}^{4}\right)}}}{c \cdot -4}}}{a \cdot 2} \]
20. Add Preprocessing

Alternative 2: 89.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 16.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (/
     (* c (* a (- -4.0)))
     (fma -2.0 b (* 2.0 (fma c (/ a b) (* (pow (* a c) 2.0) (pow b -3.0))))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 16.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((c * (a * -(-4.0))) / fma(-2.0, b, (2.0 * fma(c, (a / b), (pow((a * c), 2.0) * pow(b, -3.0)))))) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 16.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(c * Float64(a * Float64(-(-4.0)))) / fma(-2.0, b, Float64(2.0 * fma(c, Float64(a / b), Float64((Float64(a * c) ^ 2.0) * (b ^ -3.0)))))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 16.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a * (--4.0)), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * b + N[(2.0 * N[(c * N[(a / b), $MachinePrecision] + N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 16

   1. Initial program 84.0%

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

      1. sqr-neg 84.0%

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

      2. +-commutative 84.0%

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

      3. unsub-neg 84.0%

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

      4. sqr-neg 84.0%

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

      5. fma-neg 84.2%

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

      6. distribute-lft-neg-in 84.2%

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

      7. *-commutative 84.2%

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

      8. *-commutative 84.2%

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

      9. distribute-rgt-neg-in 84.2%

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

      10. metadata-eval 84.2%

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

      11. *-commutative 84.2%

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

   3. Simplified 84.2%

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

   if 16 < b

   1. Initial program 46.5%

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

      1. *-commutative 46.5%

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

   3. Simplified 46.5%

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

      1. add-cube-cbrt 44.4%

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

      2. fma-neg 44.6%

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

      3. *-commutative 44.6%

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

      4. distribute-rgt-neg-in 44.6%

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

      5. distribute-lft-neg-in 44.6%

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

      6. metadata-eval 44.6%

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

      7. *-commutative 44.6%

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

      8. cbrt-unprod 45.7%

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

      9. pow2 45.7%

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

      10. pow2 45.7%

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

      11. pow-prod-up 45.5%

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

      12. metadata-eval 45.5%

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

      13. cbrt-prod 45.1%

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

      14. pow2 45.1%

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

      15. *-commutative 45.1%

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

      16. associate-*r* 45.1%

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

   6. Applied egg-rr 45.1%

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

      1. flip-+ 45.1%

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

   8. Applied egg-rr 48.0%

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

      1. associate--r+ 99.2%

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

   10. Simplified 99.2%

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

   11. Taylor expanded in b around inf 90.5%

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

       1. fma-def 90.5%

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

       2. distribute-lft-out 90.5%

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

       3. associate-*l/ 90.5%

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

       4. *-commutative 90.5%

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

       5. unpow2 90.5%

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

       6. unpow2 90.5%

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

       7. swap-sqr 90.5%

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

       8. unpow2 90.5%

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

   13. Simplified 90.5%

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

       1. div-sub 90.5%

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

   15. Applied egg-rr 90.5%

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

       1. div-sub 90.5%

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

       2. +-inverses 90.5%

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

       3. neg-sub0 90.5%

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

       4. associate-*r* 90.5%

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

       5. *-commutative 90.5%

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

       6. distribute-neg-frac 90.5%

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

       7. associate-*r* 90.5%

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

       8. *-commutative 90.5%

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

   17. Simplified 90.5%

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

4. Final simplification 89.1%

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

Alternative 3: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * -4.0))
	return Float64(Float64(Float64(Float64((Float64(-b) ^ 2.0) - (b ^ 2.0)) - t_0) / Float64(Float64(-b) - sqrt(fma(b, b, t_0)))) / Float64(a * 2.0))
end

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

3. Simplified 54.5%

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

   1. add-cube-cbrt 52.4%

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

   2. fma-neg 52.6%

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

   3. *-commutative 52.6%

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

   4. distribute-rgt-neg-in 52.6%

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

   5. distribute-lft-neg-in 52.6%

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

   6. metadata-eval 52.6%

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

   7. *-commutative 52.6%

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

   8. cbrt-unprod 53.7%

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

   9. pow2 53.7%

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

   10. pow2 53.7%

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

   11. pow-prod-up 53.4%

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

   12. metadata-eval 53.4%

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

   13. cbrt-prod 53.1%

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

   14. pow2 53.1%

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

   15. *-commutative 53.1%

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

   16. associate-*r* 53.1%

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

6. Applied egg-rr 53.1%

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

   1. flip-+ 53.0%

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

8. Applied egg-rr 56.2%

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

   1. associate--r+ 99.2%

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

10. Simplified 99.2%

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

    1. unpow2 99.2%

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

    2. fma-def 99.2%

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

12. Applied egg-rr 99.2%

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

13. Final simplification 99.2%

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

Alternative 4: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -6 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -6e-7) t_0 (/ (- c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -6e-7) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-6d-7)) then
        tmp = t_0
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -6e-7) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -6e-7:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -6e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -6e-7)
		tmp = t_0;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -6e-7], t$95$0, N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -5.9999999999999997e-7

   1. Initial program 74.2%

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

   if -5.9999999999999997e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 26.6%

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

      1. *-commutative 26.6%

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

   3. Simplified 26.6%

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

   5. Taylor expanded in b around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. distribute-neg-frac 86.9%

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

   7. Simplified 86.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 600:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot {c}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 600.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a (pow b 3.0)) (pow c 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 600.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / pow(b, 3.0)) * pow(c, 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 600.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / (b ^ 3.0)) * (c ^ 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 600.0], 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[((-c) / b), $MachinePrecision] - N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 600

   1. Initial program 79.5%

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

      1. sqr-neg 79.5%

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

      2. +-commutative 79.5%

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

      3. unsub-neg 79.5%

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

      4. sqr-neg 79.5%

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

      5. fma-neg 79.6%

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

      6. distribute-lft-neg-in 79.6%

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

      7. *-commutative 79.6%

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

      8. *-commutative 79.6%

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

      9. distribute-rgt-neg-in 79.6%

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

      10. metadata-eval 79.6%

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

      11. *-commutative 79.6%

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

   3. Simplified 79.6%

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

   if 600 < b

   1. Initial program 40.3%

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

      1. *-commutative 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in b around inf 89.4%

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

      1. mul-1-neg 89.4%

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

      2. unsub-neg 89.4%

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

      3. mul-1-neg 89.4%

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

      4. distribute-neg-frac 89.4%

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

      5. associate-/l* 89.4%

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

      6. associate-/r/ 89.4%

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

   7. Simplified 89.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 600:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot {c}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 600:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot {c}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 600.0)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a (pow b 3.0)) (pow c 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 600.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / pow(b, 3.0)) * pow(c, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 600.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a / (b ** 3.0d0)) * (c ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 600.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / Math.pow(b, 3.0)) * Math.pow(c, 2.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 600.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a / math.pow(b, 3.0)) * math.pow(c, 2.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 600.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / (b ^ 3.0)) * (c ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 600.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - ((a / (b ^ 3.0)) * (c ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 600.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 600

   1. Initial program 79.5%

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

   if 600 < b

   1. Initial program 40.3%

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

      1. *-commutative 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in b around inf 89.4%

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

      1. mul-1-neg 89.4%

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

      2. unsub-neg 89.4%

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

      3. mul-1-neg 89.4%

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

      4. distribute-neg-frac 89.4%

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

      5. associate-/l* 89.4%

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

      6. associate-/r/ 89.4%

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

   7. Simplified 89.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 600:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot {c}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 700:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(c \cdot \frac{a}{b} + \frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{{b}^{3}}\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 700.0)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/
    (* -2.0 (+ (* c (/ a b)) (/ (* (* a c) (* a c)) (pow b 3.0))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 700.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((c * (a / b)) + (((a * c) * (a * c)) / pow(b, 3.0)))) / (a * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 700.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = ((-2.0d0) * ((c * (a / b)) + (((a * c) * (a * c)) / (b ** 3.0d0)))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 700.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((c * (a / b)) + (((a * c) * (a * c)) / Math.pow(b, 3.0)))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 700.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-2.0 * ((c * (a / b)) + (((a * c) * (a * c)) / math.pow(b, 3.0)))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 700.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64(c * Float64(a / b)) + Float64(Float64(Float64(a * c) * Float64(a * c)) / (b ^ 3.0)))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 700.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (-2.0 * ((c * (a / b)) + (((a * c) * (a * c)) / (b ^ 3.0)))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 700.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 700

   1. Initial program 79.5%

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

   if 700 < b

   1. Initial program 40.3%

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

      1. *-commutative 40.3%

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

   3. Simplified 40.3%

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

      1. add-cube-cbrt 38.6%

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

      2. fma-neg 38.6%

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

      3. *-commutative 38.6%

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

      4. distribute-rgt-neg-in 38.6%

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

      5. distribute-lft-neg-in 38.6%

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

      6. metadata-eval 38.6%

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

      7. *-commutative 38.6%

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

      8. cbrt-unprod 39.7%

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

      9. pow2 39.7%

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

      10. pow2 39.7%

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

      11. pow-prod-up 39.4%

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

      12. metadata-eval 39.4%

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

      13. cbrt-prod 39.1%

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

      14. pow2 39.1%

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

      15. *-commutative 39.1%

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

      16. associate-*r* 39.1%

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

   6. Applied egg-rr 39.1%

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

      1. flip-+ 39.1%

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

   8. Applied egg-rr 41.8%

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

      1. associate--r+ 99.2%

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

   10. Simplified 99.2%

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

   11. Taylor expanded in b around inf 89.2%

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

       1. distribute-lft-out 89.2%

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

       2. associate-*l/ 89.2%

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

       3. *-commutative 89.2%

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

       4. unpow2 89.2%

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

       5. unpow2 89.2%

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

       6. swap-sqr 89.2%

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

       7. unpow2 89.2%

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

   13. Simplified 89.2%

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

       1. unpow2 89.2%

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

   15. Applied egg-rr 89.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 700:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(c \cdot \frac{a}{b} + \frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{{b}^{3}}\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in b around inf 64.5%

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

   1. mul-1-neg 64.5%

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

   2. distribute-neg-frac 64.5%

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

7. Simplified 64.5%

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

8. Final simplification 64.5%

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

Reproduce

herbie shell --seed 2024020 
(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)))