Cubic critical, narrow range

Percentage Accurate: 55.1% → 92.1%

Time: 11.0s

Alternatives: 10

Speedup: 2.9×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b}{a} \cdot b\right) \cdot a}\\ \mathbf{if}\;b \leq 0.0265:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot b}{t\_0} + \frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}{t\_0}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \left(c \cdot c\right) \cdot \left(-1.0546875 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{0.375}{b \cdot b}\right), -0.5 \cdot c\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ b (sqrt (* (fma -3.0 c (* (/ b a) b)) a)))))
   (if (<= b 0.0265)
     (/ (+ (/ (* (- b) b) t_0) (/ (fma -3.0 (* a c) (* b b)) t_0)) (* 3.0 a))
     (/
      (fma
       (* -0.5625 (* a a))
       (* (/ (* c c) (* b b)) (/ c (* b b)))
       (fma
        a
        (*
         (* c c)
         (-
          (* -1.0546875 (/ (* (* a a) (* c c)) (pow b 6.0)))
          (/ 0.375 (* b b))))
        (* -0.5 c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b + sqrt((fma(-3.0, c, ((b / a) * b)) * a));
	double tmp;
	if (b <= 0.0265) {
		tmp = (((-b * b) / t_0) + (fma(-3.0, (a * c), (b * b)) / t_0)) / (3.0 * a);
	} else {
		tmp = fma((-0.5625 * (a * a)), (((c * c) / (b * b)) * (c / (b * b))), fma(a, ((c * c) * ((-1.0546875 * (((a * a) * (c * c)) / pow(b, 6.0))) - (0.375 / (b * b)))), (-0.5 * c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b + sqrt(Float64(fma(-3.0, c, Float64(Float64(b / a) * b)) * a)))
	tmp = 0.0
	if (b <= 0.0265)
		tmp = Float64(Float64(Float64(Float64(Float64(-b) * b) / t_0) + Float64(fma(-3.0, Float64(a * c), Float64(b * b)) / t_0)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(-0.5625 * Float64(a * a)), Float64(Float64(Float64(c * c) / Float64(b * b)) * Float64(c / Float64(b * b))), fma(a, Float64(Float64(c * c) * Float64(Float64(-1.0546875 * Float64(Float64(Float64(a * a) * Float64(c * c)) / (b ^ 6.0))) - Float64(0.375 / Float64(b * b)))), Float64(-0.5 * c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b + N[Sqrt[N[(N[(-3.0 * c + N[(N[(b / a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0265], N[(N[(N[(N[((-b) * b), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(c * c), $MachinePrecision] * N[(N[(-1.0546875 * N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0264999999999999993

   1. Initial program 86.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 87.1

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

   5. Applied rewrites 87.1%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in a around 0

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

      1. rem-square-sqrt N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if 0.0264999999999999993 < b

   1. Initial program 48.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-9}{16} \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \frac{-1}{2} \cdot c + a \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{b} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \mathsf{fma}\left(-1.0546875, \left(a \cdot a\right) \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}\right), -0.5 \cdot c\right)\right)}{b} \]

   11. Taylor expanded in c around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-9}{16} \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, {c}^{2} \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{6}} - \frac{3}{8} \cdot \frac{1}{{b}^{2}}\right), \frac{-1}{2} \cdot c\right)\right)}{b} \]
   12. Step-by-step derivation

   13. Applied rewrites 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \left(c \cdot c\right) \cdot \left(-1.0546875 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{0.375}{b \cdot b}\right), -0.5 \cdot c\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0265:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot b}{b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b}{a} \cdot b\right) \cdot a}} + \frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b}{a} \cdot b\right) \cdot a}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \left(c \cdot c\right) \cdot \left(-1.0546875 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}} - \frac{0.375}{b \cdot b}\right), -0.5 \cdot c\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c, \frac{b}{a} \cdot b\right) \cdot a\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0125:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -3.0 c (* (/ b a) b)) a)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0125)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* a 3.0))))
     (/ (fma -0.375 (/ (* a (* c c)) (* b b)) (* -0.5 c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, c, ((b / a) * b)) * a;
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0125) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(a * 3.0));
	} else {
		tmp = fma(-0.375, ((a * (c * c)) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-3.0, c, Float64(Float64(b / a) * b)) * a)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.0125)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(a * 3.0))));
	else
		tmp = Float64(fma(-0.375, Float64(Float64(a * Float64(c * c)) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c + N[(N[(b / a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0125], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(a * 3.0), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * 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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.012500000000000001

   1. Initial program 79.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 80.1%

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

   if -0.012500000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 42.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.7%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b} \]

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot {c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}}{b} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \color{blue}{\frac{a \cdot {c}^{2}}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

      10. lower-*.f64 91.5

          \[\leadsto \frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, \color{blue}{-0.5 \cdot c}\right)}{b} \]

   10. Applied rewrites 91.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0125:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3, c, \frac{b}{a} \cdot b\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b}{a} \cdot b\right) \cdot a}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b}{a} \cdot b\right) \cdot a}\\ \mathbf{if}\;b \leq 0.0265:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot b}{t\_0} + \frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}{t\_0}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ b (sqrt (* (fma -3.0 c (* (/ b a) b)) a)))))
   (if (<= b 0.0265)
     (/ (+ (/ (* (- b) b) t_0) (/ (fma -3.0 (* a c) (* b b)) t_0)) (* 3.0 a))
     (/
      (fma
       (* -0.5625 (* a a))
       (* (/ (* c c) (* b b)) (/ c (* b b)))
       (fma a (/ (* -0.375 (* c c)) (* b b)) (* -0.5 c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b + sqrt((fma(-3.0, c, ((b / a) * b)) * a));
	double tmp;
	if (b <= 0.0265) {
		tmp = (((-b * b) / t_0) + (fma(-3.0, (a * c), (b * b)) / t_0)) / (3.0 * a);
	} else {
		tmp = fma((-0.5625 * (a * a)), (((c * c) / (b * b)) * (c / (b * b))), fma(a, ((-0.375 * (c * c)) / (b * b)), (-0.5 * c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b + sqrt(Float64(fma(-3.0, c, Float64(Float64(b / a) * b)) * a)))
	tmp = 0.0
	if (b <= 0.0265)
		tmp = Float64(Float64(Float64(Float64(Float64(-b) * b) / t_0) + Float64(fma(-3.0, Float64(a * c), Float64(b * b)) / t_0)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(-0.5625 * Float64(a * a)), Float64(Float64(Float64(c * c) / Float64(b * b)) * Float64(c / Float64(b * b))), fma(a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * b)), Float64(-0.5 * c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b + N[Sqrt[N[(N[(-3.0 * c + N[(N[(b / a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0265], N[(N[(N[(N[((-b) * b), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0264999999999999993

   1. Initial program 86.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 87.1

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

   5. Applied rewrites 87.1%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in a around 0

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

      1. rem-square-sqrt N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if 0.0264999999999999993 < b

   1. Initial program 48.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-9}{16} \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \frac{-1}{2} \cdot c + a \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{b} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \mathsf{fma}\left(-1.0546875, \left(a \cdot a\right) \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}\right), -0.5 \cdot c\right)\right)}{b} \]

   11. Taylor expanded in a around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-9}{16} \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)\right)}{b} \]
   12. Step-by-step derivation

   13. Applied rewrites 92.4%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0265:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot b}{b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b}{a} \cdot b\right) \cdot a}} + \frac{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b}{a} \cdot b\right) \cdot a}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0265:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(\mathsf{fma}\left(\frac{a \cdot \frac{c}{b}}{b}, -3, 1\right) \cdot b\right) \cdot b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0265)
   (/
    (+ (- b) (sqrt (* (* (fma (/ (* a (/ c b)) b) -3.0 1.0) b) b)))
    (* 3.0 a))
   (/
    (fma
     (* -0.5625 (* a a))
     (* (/ (* c c) (* b b)) (/ c (* b b)))
     (fma a (/ (* -0.375 (* c c)) (* b b)) (* -0.5 c)))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0265) {
		tmp = (-b + sqrt(((fma(((a * (c / b)) / b), -3.0, 1.0) * b) * b))) / (3.0 * a);
	} else {
		tmp = fma((-0.5625 * (a * a)), (((c * c) / (b * b)) * (c / (b * b))), fma(a, ((-0.375 * (c * c)) / (b * b)), (-0.5 * c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0265)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(fma(Float64(Float64(a * Float64(c / b)) / b), -3.0, 1.0) * b) * b))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(-0.5625 * Float64(a * a)), Float64(Float64(Float64(c * c) / Float64(b * b)) * Float64(c / Float64(b * b))), fma(a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * b)), Float64(-0.5 * c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0265], N[(N[((-b) + N[Sqrt[N[(N[(N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0264999999999999993

   1. Initial program 86.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if 0.0264999999999999993 < b

   1. Initial program 48.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-9}{16} \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \frac{-1}{2} \cdot c + a \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{b} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.3%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \mathsf{fma}\left(-1.0546875, \left(a \cdot a\right) \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}\right), -0.5 \cdot c\right)\right)}{b} \]

   11. Taylor expanded in a around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-9}{16} \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)\right)}{b} \]
   12. Step-by-step derivation

   13. Applied rewrites 92.4%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0125:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0125)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (/ (fma -0.375 (/ (* a (* c c)) (* b b)) (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0125) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = fma(-0.375, ((a * (c * c)) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.0125)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-0.375, Float64(Float64(a * Float64(c * c)) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0125], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * 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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.012500000000000001

   1. Initial program 79.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 80.0

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

   4. Applied rewrites 80.0%

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

   if -0.012500000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 42.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.7%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b} \]

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot {c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}}{b} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \color{blue}{\frac{a \cdot {c}^{2}}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

      10. lower-*.f64 91.5

          \[\leadsto \frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, \color{blue}{-0.5 \cdot c}\right)}{b} \]

   10. Applied rewrites 91.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0125:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375, c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0125)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (* (fma (* (/ a (* (* b b) b)) -0.375) c (/ -0.5 b)) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0125) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = fma(((a / ((b * b) * b)) * -0.375), c, (-0.5 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.0125)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(a / Float64(Float64(b * b) * b)) * -0.375), c, Float64(-0.5 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0125], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] * c + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * c), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.012500000000000001

   1. Initial program 79.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 80.0

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

   4. Applied rewrites 80.0%

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

   if -0.012500000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 42.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right)} \cdot c \]

      3. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

      5. associate-*l/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

      6. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right)} \cdot c \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{{b}^{3}} \cdot \frac{-3}{8}}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{{b}^{3}} \cdot \frac{-3}{8}}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{{b}^{3}}} \cdot \frac{-3}{8}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{a}{\color{blue}{{b}^{3}}} \cdot \frac{-3}{8}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot \frac{-3}{8}, c, \color{blue}{\frac{-1}{2}} \cdot \frac{1}{b}\right) \cdot c \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot \frac{-3}{8}, c, \color{blue}{\frac{\frac{-1}{2} \cdot 1}{b}}\right) \cdot c \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot \frac{-3}{8}, c, \frac{\color{blue}{\frac{-1}{2}}}{b}\right) \cdot c \]

      16. lower-/.f64 91.3

          \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot -0.375, c, \color{blue}{\frac{-0.5}{b}}\right) \cdot c \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot -0.375, c, \frac{-0.5}{b}\right) \cdot c} \]
   6. Step-by-step derivation

   7. Applied rewrites 91.3%

      \[\leadsto \mathsf{fma}\left(\frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375, c, \frac{-0.5}{b}\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0125:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{b \cdot b}\right) - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0125)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (/ (* c (- (* -0.375 (* a (/ c (* b b)))) 0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0125) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (c * ((-0.375 * (a * (c / (b * b)))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.0125)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / Float64(b * b)))) - 0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0125], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(-0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.012500000000000001

   1. Initial program 79.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 80.0

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

   4. Applied rewrites 80.0%

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

   if -0.012500000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 42.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.7%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b} \]

   8. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.3%

       \[\leadsto \frac{c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{b \cdot b}\right) - 0.5\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.375, a \cdot \frac{c}{b \cdot b}, 0.5\right)}{-b} \cdot c \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ (fma 0.375 (* a (/ c (* b b))) 0.5) (- b)) c))

C

double code(double a, double b, double c) {
	return (fma(0.375, (a * (c / (b * b))), 0.5) / -b) * c;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(0.375, Float64(a * Float64(c / Float64(b * b))), 0.5) / Float64(-b)) * c)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / (-b)), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   2. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right)} \cdot c \]

   3. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   4. associate-*r* N/A

      \[\leadsto \left(\frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   5. associate-*l/ N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   6. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right)} \cdot c \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{{b}^{3}} \cdot \frac{-3}{8}}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{{b}^{3}} \cdot \frac{-3}{8}}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   11. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{{b}^{3}}} \cdot \frac{-3}{8}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   12. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{a}{\color{blue}{{b}^{3}}} \cdot \frac{-3}{8}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot \frac{-3}{8}, c, \color{blue}{\frac{-1}{2}} \cdot \frac{1}{b}\right) \cdot c \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot \frac{-3}{8}, c, \color{blue}{\frac{\frac{-1}{2} \cdot 1}{b}}\right) \cdot c \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot \frac{-3}{8}, c, \frac{\color{blue}{\frac{-1}{2}}}{b}\right) \cdot c \]

   16. lower-/.f64 83.5

       \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot -0.375, c, \color{blue}{\frac{-0.5}{b}}\right) \cdot c \]

5. Applied rewrites 83.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot -0.375, c, \frac{-0.5}{b}\right) \cdot c} \]

6. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{-1}{2}}{b} \cdot c \]
7. Step-by-step derivation

8. Applied rewrites 67.0%

   \[\leadsto \frac{-0.5}{b} \cdot c \]

9. Taylor expanded in b around -inf

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

11. Applied rewrites 83.5%

    \[\leadsto \left(-\frac{\mathsf{fma}\left(0.375, a \cdot \frac{c}{b \cdot b}, 0.5\right)}{b}\right) \cdot c \]

12. Final simplification 83.5%

    \[\leadsto \frac{\mathsf{fma}\left(0.375, a \cdot \frac{c}{b \cdot b}, 0.5\right)}{-b} \cdot c \]
13. Add Preprocessing

Alternative 9: 64.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) -0.5))

C

double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

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) * (-0.5d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Python

def code(a, b, c):
	return (c / b) * -0.5

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * -0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * -0.5;
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 67.1

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

5. Applied rewrites 67.1%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Add Preprocessing

Alternative 10: 64.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{b} \cdot c \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ -0.5 b) c))

C

double code(double a, double b, double c) {
	return (-0.5 / b) * c;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) / b) * c
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 / b) * c;
}

Python

def code(a, b, c):
	return (-0.5 / b) * c

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 / b) * c)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 / b) * c;
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 / b), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   2. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right)} \cdot c \]

   3. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   4. associate-*r* N/A

      \[\leadsto \left(\frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   5. associate-*l/ N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   6. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right)} \cdot c \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{{b}^{3}} \cdot \frac{-3}{8}}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{{b}^{3}} \cdot \frac{-3}{8}}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   11. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{{b}^{3}}} \cdot \frac{-3}{8}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   12. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{a}{\color{blue}{{b}^{3}}} \cdot \frac{-3}{8}, c, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b}\right) \cdot c \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot \frac{-3}{8}, c, \color{blue}{\frac{-1}{2}} \cdot \frac{1}{b}\right) \cdot c \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot \frac{-3}{8}, c, \color{blue}{\frac{\frac{-1}{2} \cdot 1}{b}}\right) \cdot c \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot \frac{-3}{8}, c, \frac{\color{blue}{\frac{-1}{2}}}{b}\right) \cdot c \]

   16. lower-/.f64 83.5

       \[\leadsto \mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot -0.375, c, \color{blue}{\frac{-0.5}{b}}\right) \cdot c \]

5. Applied rewrites 83.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{{b}^{3}} \cdot -0.375, c, \frac{-0.5}{b}\right) \cdot c} \]

6. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{-1}{2}}{b} \cdot c \]
7. Step-by-step derivation

8. Applied rewrites 67.0%

   \[\leadsto \frac{-0.5}{b} \cdot c \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))