Quadratic roots, medium range

Percentage Accurate: 31.8% → 99.6%

Time: 9.9s

Alternatives: 9

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: 31.8% 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: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 34.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 34.1

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

4. Applied rewrites 34.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

6. Applied rewrites 35.3%

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

7. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.3

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

9. Applied rewrites 99.3%

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 34.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 34.1

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

4. Applied rewrites 34.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. flip-+ N/A

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

   5. lower-/.f64 N/A

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

6. Applied rewrites 33.5%

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

7. Taylor expanded in a around 0

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

   1. lower-/.f64 99.4

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

9. Applied rewrites 99.4%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-/r/ N/A

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

    4. lift-*.f64 N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. lower-*.f64 N/A

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

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 34.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 34.1

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

4. Applied rewrites 34.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

6. Applied rewrites 35.3%

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

7. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.3

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

9. Applied rewrites 99.3%

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

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 99.3

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lift-*.f64 99.3

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

11. Applied rewrites 99.3%

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

Alternative 4: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 34.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 34.1

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

4. Applied rewrites 34.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

6. Applied rewrites 35.3%

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

7. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.3

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

9. Applied rewrites 99.3%

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

Alternative 5: 90.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00078:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{a}{b \cdot b}, \frac{c \cdot c}{b}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.00078)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (- (fma (/ a (* b b)) (/ (* c c) b) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.00078) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = -fma((a / (b * b)), ((c * c) / b), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.00078)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(-fma(Float64(a / Float64(b * b)), Float64(Float64(c * c) / b), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.00078], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], (-N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999986e-4

   1. Initial program 78.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 78.4

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

   4. Applied rewrites 78.4%

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

   if 7.79999999999999986e-4 < b

   1. Initial program 30.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 92.3

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

   8. Applied rewrites 92.3%

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

   10. Applied rewrites 92.3%

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

Alternative 6: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00078:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{a}}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.00078)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/ (/ c a) (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.00078) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (c / a) / fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.00078)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c / a) / fma(Float64(b / a), -1.0, Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.00078], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / a), $MachinePrecision] / N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999986e-4

   1. Initial program 78.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 78.4

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

   4. Applied rewrites 78.4%

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

   if 7.79999999999999986e-4 < b

   1. Initial program 30.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 30.0

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

   4. Applied rewrites 30.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. flip-+ N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 29.4%

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

   7. Taylor expanded in a around 0

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

      1. lower-/.f64 99.4

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

   9. Applied rewrites 99.4%

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

   10. Taylor expanded in c around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 92.3

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

   12. Applied rewrites 92.3%

       \[\leadsto \frac{\frac{c}{a}}{\color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00078:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot c\right) \cdot a}{\left(-b\right) \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.00078)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/ (- (/ (* (* c c) a) (* (- b) b)) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.00078) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = ((((c * c) * a) / (-b * b)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.00078)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(c * c) * a) / Float64(Float64(-b) * b)) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.00078], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[((-b) * b), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999986e-4

   1. Initial program 78.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 78.4

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

   4. Applied rewrites 78.4%

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

   if 7.79999999999999986e-4 < b

   1. Initial program 30.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 92.3

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

   8. Applied rewrites 92.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 92.3%

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

Alternative 8: 90.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.1%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 95.3%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-out N/A

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

   5. lower-neg.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-/.f64 89.5

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

8. Applied rewrites 89.5%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 89.4%

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

Alternative 9: 81.1% 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 34.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 79.3

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

5. Applied rewrites 79.3%

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

Reproduce

herbie shell --seed 2024337 
(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)))