Quadratic roots, medium range

Percentage Accurate: 31.6% → 99.4%

Time: 11.9s

Alternatives: 8

Speedup: 3.6×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Initial Program: 31.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c, \frac{b}{a} \cdot b\right) \cdot a}\right)}^{-1} \cdot \frac{c \cdot \left(a \cdot 4\right)}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (pow (- (- b) (sqrt (* (fma -4.0 c (* (/ b a) b)) a))) -1.0)
  (/ (* c (* a 4.0)) (* 2.0 a))))

C

double code(double a, double b, double c) {
	return pow((-b - sqrt((fma(-4.0, c, ((b / a) * b)) * a))), -1.0) * ((c * (a * 4.0)) / (2.0 * a));
}

Julia

function code(a, b, c)
	return Float64((Float64(Float64(-b) - sqrt(Float64(fma(-4.0, c, Float64(Float64(b / a) * b)) * a))) ^ -1.0) * Float64(Float64(c * Float64(a * 4.0)) / Float64(2.0 * a)))
end

Wolfram

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

Derivation

1. Initial program 32.4%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 32.3

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

5. Applied rewrites 32.3%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 32.9%

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

8. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.3

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

10. Applied rewrites 99.3%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. associate-/r/ N/A

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

12. Applied rewrites 99.4%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\frac{b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{4 \cdot \left(a \cdot c\right)}\right)}^{-1}}{2 \cdot \left(-a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (pow (/ (+ b (sqrt (fma -4.0 (* c a) (* b b)))) (* 4.0 (* a c))) -1.0)
  (* 2.0 (- a))))

C

double code(double a, double b, double c) {
	return pow(((b + sqrt(fma(-4.0, (c * a), (b * b)))) / (4.0 * (a * c))), -1.0) / (2.0 * -a);
}

Julia

function code(a, b, c)
	return Float64((Float64(Float64(b + sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))) / Float64(4.0 * Float64(a * c))) ^ -1.0) / Float64(2.0 * Float64(-a)))
end

Wolfram

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

Derivation

1. Initial program 32.4%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 32.3

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

5. Applied rewrites 32.3%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 32.9%

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

8. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.3

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

10. Applied rewrites 99.3%

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

11. Taylor expanded in b around 0

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

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

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

    2. metadata-eval N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 99.4

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

13. Applied rewrites 99.4%

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

14. Final simplification 99.4%

    \[\leadsto \frac{{\left(\frac{b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{4 \cdot \left(a \cdot c\right)}\right)}^{-1}}{2 \cdot \left(-a\right)} \]
15. Add Preprocessing

Alternative 3: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}\right)}^{-1}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (pow (/ (fma -0.5 (/ b a) (* 0.5 (/ c b))) c) -1.0) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return pow((fma(-0.5, (b / a), (0.5 * (c / b))) / c), -1.0) / (2.0 * a);
}

Julia

function code(a, b, c)
	return Float64((Float64(fma(-0.5, Float64(b / a), Float64(0.5 * Float64(c / b))) / c) ^ -1.0) / Float64(2.0 * a))
end

Wolfram

code[a_, b_, c_] := N[(N[Power[N[(N[(-0.5 * N[(b / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.4%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 32.3

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

5. Applied rewrites 32.3%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 32.9%

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

8. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 89.7

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

10. Applied rewrites 89.7%

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

11. Final simplification 89.7%

    \[\leadsto \frac{{\left(\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}\right)}^{-1}}{2 \cdot a} \]
12. Add Preprocessing

Alternative 4: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.4%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 32.3

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

5. Applied rewrites 32.3%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 32.9%

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

8. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.3

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

10. Applied rewrites 99.3%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. clear-num N/A

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

    4. lower-/.f64 99.4

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

12. Applied rewrites 99.4%

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

13. Final simplification 99.4%

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

Alternative 5: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(a \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(-4, c, \frac{b}{a} \cdot b\right) \cdot a}} \cdot \frac{0.5}{-a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ (* c (* a 4.0)) (+ b (sqrt (* (fma -4.0 c (* (/ b a) b)) a))))
  (/ 0.5 (- a))))

C

double code(double a, double b, double c) {
	return ((c * (a * 4.0)) / (b + sqrt((fma(-4.0, c, ((b / a) * b)) * a)))) * (0.5 / -a);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(a * 4.0)) / Float64(b + sqrt(Float64(fma(-4.0, c, Float64(Float64(b / a) * b)) * a)))) * Float64(0.5 / Float64(-a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(-4.0 * c + N[(N[(b / a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.4%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 32.3

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

5. Applied rewrites 32.3%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 32.9%

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

8. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.3

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

10. Applied rewrites 99.3%

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

    1. lift-/.f64 N/A

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

    2. div-inv N/A

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

    3. lower-*.f64 N/A

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

12. Applied rewrites 99.3%

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

13. Final simplification 99.3%

    \[\leadsto \frac{c \cdot \left(a \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(-4, c, \frac{b}{a} \cdot b\right) \cdot a}} \cdot \frac{0.5}{-a} \]
14. Add Preprocessing

Alternative 6: 90.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.4%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 94.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-5 \cdot \left(a \cdot {c}^{4}\right) + -2 \cdot \left({b}^{2} \cdot {c}^{3}\right)}{{b}^{7}}, a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
7. Step-by-step derivation

8. Applied rewrites 94.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot a, {c}^{4}, \left(-2 \cdot \left(b \cdot b\right)\right) \cdot {c}^{3}\right)}{{b}^{7}}, a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]

9. Taylor expanded in b around -inf

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

11. Applied rewrites 89.6%

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

12. Final simplification 89.6%

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

Alternative 7: 90.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-1 - \frac{c \cdot a}{b \cdot b}}{b} \cdot c \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return ((-1.0 - ((c * a) / (b * b))) / 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 = (((-1.0d0) - ((c * a) / (b * b))) / b) * c
end function

Java

public static double code(double a, double b, double c) {
	return ((-1.0 - ((c * a) / (b * b))) / b) * c;
}

Python

def code(a, b, c):
	return ((-1.0 - ((c * a) / (b * b))) / b) * c

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((-1.0 - ((c * a) / (b * b))) / b) * c;
end

Wolfram

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

Derivation

1. Initial program 32.4%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. distribute-neg-frac N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r* N/A

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

   7. associate-*l/ N/A

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

   8. associate-*r/ N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 89.5%

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

7. Applied rewrites 89.5%

   \[\leadsto \frac{-1 - \frac{c \cdot a}{b \cdot b}}{b} \cdot c \]
8. Add Preprocessing

Alternative 8: 81.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.4%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

   4. lower-neg.f64 80.5

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

5. Applied rewrites 80.5%

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

Reproduce

herbie shell --seed 2024308 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))