Cubic critical, medium range

Percentage Accurate: 31.6% → 99.0%

Time: 10.8s

Alternatives: 5

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.6% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(0.3333333333333333 \cdot \frac{c \cdot \left(3 \cdot a\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}\right) \cdot \frac{-1}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (*
   0.3333333333333333
   (/ (* c (* 3.0 a)) (+ (sqrt (fma (* -3.0 c) a (* b b))) b)))
  (/ -1.0 a)))

C

double code(double a, double b, double c) {
	return (0.3333333333333333 * ((c * (3.0 * a)) / (sqrt(fma((-3.0 * c), a, (b * b))) + b))) * (-1.0 / a);
}

Julia

function code(a, b, c)
	return Float64(Float64(0.3333333333333333 * Float64(Float64(c * Float64(3.0 * a)) / Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) + b))) * Float64(-1.0 / a))
end

Wolfram

code[a_, b_, c_] := N[(N[(0.3333333333333333 * N[(N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.2%

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

4. Applied rewrites 31.2%

   \[\leadsto \color{blue}{\frac{-1}{a} \cdot \left(\left(b - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right) \cdot 0.3333333333333333\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 31.3

      \[\leadsto \frac{-1}{a} \cdot \left(\left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot -3}\right)}\right) \cdot 0.3333333333333333\right) \]

6. Applied rewrites 31.3%

   \[\leadsto \frac{-1}{a} \cdot \left(\left(b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}}\right) \cdot 0.3333333333333333\right) \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 31.8%

   \[\leadsto \frac{-1}{a} \cdot \left(\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}} \cdot 0.3333333333333333\right) \]

9. Taylor expanded in c around 0

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

    1. associate-*r* N/A

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

    2. lower-*.f64 N/A

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

    3. lower-*.f64 99.0

       \[\leadsto \frac{-1}{a} \cdot \left(\frac{\color{blue}{\left(3 \cdot a\right)} \cdot c}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b} \cdot 0.3333333333333333\right) \]

11. Applied rewrites 99.0%

    \[\leadsto \frac{-1}{a} \cdot \left(\frac{\color{blue}{\left(3 \cdot a\right) \cdot c}}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b} \cdot 0.3333333333333333\right) \]

12. Final simplification 99.0%

    \[\leadsto \left(0.3333333333333333 \cdot \frac{c \cdot \left(3 \cdot a\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}\right) \cdot \frac{-1}{a} \]
13. Add Preprocessing

Alternative 2: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(-2, b, \left(\frac{c}{b} \cdot a\right) \cdot 1.5\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (fma -2.0 b (* (* (/ c b) a) 1.5)) c)))

C

double code(double a, double b, double c) {
	return 1.0 / (fma(-2.0, b, (((c / b) * a) * 1.5)) / c);
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(fma(-2.0, b, Float64(Float64(Float64(c / b) * a) * 1.5)) / c))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * b + N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.2%

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

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

   13. *-commutative N/A

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

   14. lower-*.f64 31.2

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

5. Applied rewrites 31.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 31.2

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-neg.f64 N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 31.2

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

7. Applied rewrites 31.2%

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

8. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 90.6

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \left(a \cdot \color{blue}{\frac{c}{b}}\right)\right)}{c}} \]

10. Applied rewrites 90.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{c}}} \]

11. Final simplification 90.6%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, \left(\frac{c}{b} \cdot a\right) \cdot 1.5\right)}{c}} \]
12. Add Preprocessing

Alternative 3: 91.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\frac{b}{c}, -2, \frac{a}{b} \cdot 1.5\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (fma (/ b c) -2.0 (* (/ a b) 1.5))))

C

double code(double a, double b, double c) {
	return 1.0 / fma((b / c), -2.0, ((a / b) * 1.5));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(Float64(b / c), -2.0, Float64(Float64(a / b) * 1.5)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(b / c), $MachinePrecision] * -2.0 + N[(N[(a / b), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.2%

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

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

   13. *-commutative N/A

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

   14. lower-*.f64 31.2

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

5. Applied rewrites 31.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 31.2

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-neg.f64 N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 31.2

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

7. Applied rewrites 31.2%

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

8. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 90.5

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{b}{c}, -2, 1.5 \cdot \color{blue}{\frac{a}{b}}\right)} \]

10. Applied rewrites 90.5%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{b}{c}, -2, 1.5 \cdot \frac{a}{b}\right)}} \]

11. Final simplification 90.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{b}{c}, -2, \frac{a}{b} \cdot 1.5\right)} \]
12. Add Preprocessing

Alternative 4: 81.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
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 81.3

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

5. Applied rewrites 81.3%

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

6. Final simplification 81.3%

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

Alternative 5: 81.0% 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 31.2%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
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 81.3

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

5. Applied rewrites 81.3%

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

7. Applied rewrites 81.0%

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

8. Final simplification 81.0%

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

Reproduce

herbie shell --seed 2024278 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))