Quadratic roots, narrow range

Percentage Accurate: 55.5% → 99.4%

Time: 10.8s

Alternatives: 10

Speedup: 3.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

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: 55.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 54.2

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

5. Applied rewrites 54.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

7. Applied rewrites 55.2%

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

8. Taylor expanded in a around 0

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

   1. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

Alternative 2: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 54.2

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

5. Applied rewrites 54.2%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 55.2%

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

8. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.2

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

10. Applied rewrites 99.2%

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

11. Final simplification 99.2%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 54.2

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

5. Applied rewrites 54.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

7. Applied rewrites 55.2%

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

8. Taylor expanded in a around 0

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

   1. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

11. Taylor expanded in a around 0

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

13. Applied rewrites 99.4%

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

Alternative 4: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{t\_0} + b}{t\_0 - b \cdot b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 37.0)
     (/ 0.5 (* (/ (+ (sqrt t_0) b) (- t_0 (* b b))) a))
     (/ 0.5 (fma (/ a b) 0.5 (* (/ b c) -0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 37.0) {
		tmp = 0.5 / (((sqrt(t_0) + b) / (t_0 - (b * b))) * a);
	} else {
		tmp = 0.5 / fma((a / b), 0.5, ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(0.5 / Float64(Float64(Float64(sqrt(t_0) + b) / Float64(t_0 - Float64(b * b))) * a));
	else
		tmp = Float64(0.5 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 37.0], N[(0.5 / N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. frac-times N/A

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

   4. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\frac{0.5}{{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)}^{-1} \cdot a}} \]

   6. Applied rewrites 84.7%

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

   if 37 < b

   1. Initial program 43.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. frac-times N/A

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

   4. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\frac{0.5}{{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)}^{-1} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 5: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 c) a)))) (* 2.0 a))
   (/ 0.5 (fma (/ a b) 0.5 (* (/ b c) -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * c) * a)))) / (2.0 * a);
	} else {
		tmp = 0.5 / fma((a / b), 0.5, ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a)))) / Float64(2.0 * a));
	else
		tmp = Float64(0.5 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 83.4

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

   4. Applied rewrites 83.4%

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

   if 37 < b

   1. Initial program 43.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. frac-times N/A

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

   4. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\frac{0.5}{{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)}^{-1} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 6: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/ (- (sqrt (fma (* -4.0 c) a (* b b))) b) (* 2.0 a))
   (/ 0.5 (fma (/ a b) 0.5 (* (/ b c) -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = 0.5 / fma((a / b), 0.5, ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(0.5 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 83.1

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 83.1

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

   4. Applied rewrites 83.1%

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

   if 37 < b

   1. Initial program 43.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. frac-times N/A

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

   4. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\frac{0.5}{{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)}^{-1} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 7: 81.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. frac-times N/A

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

4. Applied rewrites 54.2%

   \[\leadsto \color{blue}{\frac{0.5}{{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)}^{-1} \cdot a}} \]

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 82.3

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

7. Applied rewrites 82.3%

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

Alternative 8: 81.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 91.8%

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

6. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r/ N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 81.6

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

8. Applied rewrites 81.6%

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

9. Final simplification 81.6%

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

Alternative 9: 64.3% 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 54.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 64.7

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

5. Applied rewrites 64.7%

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

Alternative 10: 1.6% accurate, 4.2× 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(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 54.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 64.7

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

5. Applied rewrites 64.7%

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

7. Applied rewrites 64.7%

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

9. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2024319 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))