Cubic critical, narrow range

Percentage Accurate: 55.6% → 90.6%

Time: 10.0s

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: 55.6% 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: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \left(\mathsf{fma}\left(a \cdot c, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/
   (fma
    (* -1.0546875 (pow c 4.0))
    (* a a)
    (* (* (fma (* a c) -0.5625 (* (* b b) -0.375)) (* c c)) (* b b)))
   (pow b 7.0))
  a
  (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	return fma((fma((-1.0546875 * pow(c, 4.0)), (a * a), ((fma((a * c), -0.5625, ((b * b) * -0.375)) * (c * c)) * (b * b))) / pow(b, 7.0)), a, ((c / b) * -0.5));
}

Julia

function code(a, b, c)
	return fma(Float64(fma(Float64(-1.0546875 * (c ^ 4.0)), Float64(a * a), Float64(Float64(fma(Float64(a * c), -0.5625, Float64(Float64(b * b) * -0.375)) * Float64(c * c)) * Float64(b * b))) / (b ^ 7.0)), a, Float64(Float64(c / b) * -0.5))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-1.0546875 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(N[(a * c), $MachinePrecision] * -0.5625 + N[(N[(b * b), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

5. Applied rewrites 91.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
7. Step-by-step derivation

8. Applied rewrites 91.1%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \mathsf{fma}\left(\left(-0.375 \cdot \left(b \cdot b\right)\right) \cdot c, c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128} \cdot {c}^{4}, a \cdot a, \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
10. Step-by-step derivation

11. Applied rewrites 91.1%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \left(\mathsf{fma}\left(a \cdot c, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

12. Final simplification 91.1%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \left(\mathsf{fma}\left(a \cdot c, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]
13. Add Preprocessing

Alternative 2: 88.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;b \leq 75:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{5}}, -0.5625, \frac{-0.375}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -3.0 c (/ (* b b) a)) a)))
   (if (<= b 75.0)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* a 3.0))))
     (fma
      (* (fma (* a (/ c (pow b 5.0))) -0.5625 (/ -0.375 (pow b 3.0))) (* c c))
      a
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, c, ((b * b) / a)) * a;
	double tmp;
	if (b <= 75.0) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(a * 3.0));
	} else {
		tmp = fma((fma((a * (c / pow(b, 5.0))), -0.5625, (-0.375 / pow(b, 3.0))) * (c * c)), a, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-3.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (b <= 75.0)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(a * 3.0))));
	else
		tmp = fma(Float64(fma(Float64(a * Float64(c / (b ^ 5.0))), -0.5625, Float64(-0.375 / (b ^ 3.0))) * Float64(c * c)), a, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[b, 75.0], 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[(N[(N[(a * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 75

   1. Initial program 81.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 82.1%

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

   if 75 < b

   1. Initial program 49.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 94.9%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \mathsf{fma}\left(\left(-0.375 \cdot \left(b \cdot b\right)\right) \cdot c, c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128} \cdot {c}^{4}, a \cdot a, \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.9%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \left(\mathsf{fma}\left(a \cdot c, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

   12. Taylor expanded in c around 0

       \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{5}} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right), a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 92.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{5}}, -0.5625, \frac{-0.375}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 75:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{5}}, -0.5625, \frac{-0.375}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;b \leq 75:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -3.0 c (/ (* b b) a)) a)))
   (if (<= b 75.0)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* a 3.0))))
     (*
      (fma
       (fma (* c -0.5625) (* a (/ a (pow b 5.0))) (* (/ a (pow b 3.0)) -0.375))
       c
       (/ -0.5 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, c, ((b * b) / a)) * a;
	double tmp;
	if (b <= 75.0) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(a * 3.0));
	} else {
		tmp = fma(fma((c * -0.5625), (a * (a / pow(b, 5.0))), ((a / pow(b, 3.0)) * -0.375)), c, (-0.5 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-3.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (b <= 75.0)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(a * 3.0))));
	else
		tmp = Float64(fma(fma(Float64(c * -0.5625), Float64(a * Float64(a / (b ^ 5.0))), Float64(Float64(a / (b ^ 3.0)) * -0.375)), c, Float64(-0.5 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[b, 75.0], 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[(N[(N[(c * -0.5625), $MachinePrecision] * N[(a * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * c + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 75

   1. Initial program 81.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 82.1%

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

   if 75 < b

   1. Initial program 49.1%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   5. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 75:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;b \leq 75:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{{b}^{3}}, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -3.0 c (/ (* b b) a)) a)))
   (if (<= b 75.0)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* a 3.0))))
     (fma (* -0.375 a) (* c (/ c (pow b 3.0))) (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, c, ((b * b) / a)) * a;
	double tmp;
	if (b <= 75.0) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(a * 3.0));
	} else {
		tmp = fma((-0.375 * a), (c * (c / pow(b, 3.0))), ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-3.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (b <= 75.0)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(a * 3.0))));
	else
		tmp = fma(Float64(-0.375 * a), Float64(c * Float64(c / (b ^ 3.0))), Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[b, 75.0], 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 * a), $MachinePrecision] * N[(c * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 75

   1. Initial program 81.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 82.1%

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

   if 75 < b

   1. Initial program 49.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 88.0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 75:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{{b}^{3}}, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;b \leq 75:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -3.0 c (/ (* b b) a)) a)))
   (if (<= b 75.0)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* a 3.0))))
     (/ (fma (/ (* -0.375 a) b) (/ (* c c) b) (* -0.5 c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, c, ((b * b) / a)) * a;
	double tmp;
	if (b <= 75.0) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(a * 3.0));
	} else {
		tmp = fma(((-0.375 * a) / b), ((c * c) / b), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-3.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (b <= 75.0)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(a * 3.0))));
	else
		tmp = Float64(fma(Float64(Float64(-0.375 * a) / b), Float64(Float64(c * c) / 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 * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[b, 75.0], 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[(N[(N[(-0.375 * a), $MachinePrecision] / b), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 75

   1. Initial program 81.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 82.1%

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

   if 75 < b

   1. Initial program 49.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   7. 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. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.8

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

   8. Applied rewrites 87.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 75:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.2e-6)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.2e-6) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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)) <= -1.2e-6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	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], -1.2e-6], 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 / b), $MachinePrecision] * -0.5), $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)) < -1.1999999999999999e-6

   1. Initial program 73.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 73.6

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

   4. Applied rewrites 73.6%

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

   if -1.1999999999999999e-6 < (/.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 33.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 82.4

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

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 75:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 75.0)
   (/ (+ (- b) (sqrt (fma (* -3.0 a) c (* b b)))) (* 3.0 a))
   (/ (fma (/ (* -0.375 a) b) (/ (* c c) b) (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 75.0) {
		tmp = (-b + sqrt(fma((-3.0 * a), c, (b * b)))) / (3.0 * a);
	} else {
		tmp = fma(((-0.375 * a) / b), ((c * c) / b), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 75.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(-0.375 * a) / b), Float64(Float64(c * c) / b), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 75.0], N[(N[((-b) + N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 * a), $MachinePrecision] / b), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 75

   1. Initial program 81.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 81.4

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

   4. Applied rewrites 81.4%

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

   if 75 < b

   1. Initial program 49.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   7. 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. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.8

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

   8. Applied rewrites 87.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 75:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% 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 57.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 63.1

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

5. Applied rewrites 63.1%

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

6. Final simplification 63.1%

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

Alternative 9: 64.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 * ((-0.5d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 63.1

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

5. Applied rewrites 63.1%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 63.1%

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

8. Final simplification 63.1%

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

Alternative 10: 3.2% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 57.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 56.8

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

5. Applied rewrites 56.8%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

7. Applied rewrites 56.9%

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

8. Taylor expanded in a around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-inverses N/A

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

   4. div0 3.2

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

10. Applied rewrites 3.2%

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

Reproduce

herbie shell --seed 2024326 
(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)))