Quadratic roots, medium range

Percentage Accurate: 31.9% → 95.4%

Time: 12.5s

Alternatives: 10

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: 31.9% 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: 95.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c}{b \cdot b} \cdot \frac{c \cdot c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.25}{a}, \frac{20}{{b}^{6}} \cdot \left({a}^{4} \cdot {c}^{4}\right), -\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (* (* -2.0 a) a)
   (* (/ c (* b b)) (/ (* c c) (* b b)))
   (fma
    (/ -0.25 a)
    (* (/ 20.0 (pow b 6.0)) (* (pow a 4.0) (pow c 4.0)))
    (- (fma (/ c b) (/ (* c a) b) c))))
  b))

C

double code(double a, double b, double c) {
	return fma(((-2.0 * a) * a), ((c / (b * b)) * ((c * c) / (b * b))), fma((-0.25 / a), ((20.0 / pow(b, 6.0)) * (pow(a, 4.0) * pow(c, 4.0))), -fma((c / b), ((c * a) / b), c))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(-2.0 * a) * a), Float64(Float64(c / Float64(b * b)) * Float64(Float64(c * c) / Float64(b * b))), fma(Float64(-0.25 / a), Float64(Float64(20.0 / (b ^ 6.0)) * Float64((a ^ 4.0) * (c ^ 4.0))), Float64(-fma(Float64(c / b), Float64(Float64(c * a) / b), c)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 / a), $MachinePrecision] * N[(N[(20.0 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 97.4%

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

7. Applied rewrites 97.4%

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

8. Final simplification 97.4%

   \[\leadsto \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c}{b \cdot b} \cdot \frac{c \cdot c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.25}{a}, \frac{20}{{b}^{6}} \cdot \left({a}^{4} \cdot {c}^{4}\right), -\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)\right)\right)}{b} \]
9. Add Preprocessing

Alternative 2: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot a\right) \cdot c\\ \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c}{b \cdot b} \cdot \frac{c \cdot c}{b \cdot b}, \mathsf{fma}\left(-5 \cdot \frac{t\_0 \cdot t\_0}{{b}^{6}} - \frac{c}{b} \cdot \frac{c}{b}, a, -c\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c a) c)))
   (/
    (fma
     (* (* -2.0 a) a)
     (* (/ c (* b b)) (/ (* c c) (* b b)))
     (fma
      (- (* -5.0 (/ (* t_0 t_0) (pow b 6.0))) (* (/ c b) (/ c b)))
      a
      (- c)))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * c;
	return fma(((-2.0 * a) * a), ((c / (b * b)) * ((c * c) / (b * b))), fma(((-5.0 * ((t_0 * t_0) / pow(b, 6.0))) - ((c / b) * (c / b))), a, -c)) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * c)
	return Float64(fma(Float64(Float64(-2.0 * a) * a), Float64(Float64(c / Float64(b * b)) * Float64(Float64(c * c) / Float64(b * b))), fma(Float64(Float64(-5.0 * Float64(Float64(t_0 * t_0) / (b ^ 6.0))) - Float64(Float64(c / b) * Float64(c / b))), a, Float64(-c))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * c), $MachinePrecision]}, N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-5.0 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + (-c)), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 97.4%

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

7. Applied rewrites 97.4%

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

8. Taylor expanded in a around 0

   \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot -2\right) \cdot a, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, a \cdot \left(-5 \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} - \frac{{c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]
9. Step-by-step derivation

10. Applied rewrites 97.4%

    \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot -2\right) \cdot a, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{{c}^{4} \cdot \left(a \cdot a\right)}{{b}^{6}} \cdot -5 - \frac{c}{b} \cdot \frac{c}{b}, a, -c\right)\right)}{b} \]
11. Step-by-step derivation

12. Applied rewrites 97.4%

    \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot -2\right) \cdot a, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{\left(\left(a \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot c\right) \cdot c\right)}{{b}^{6}} \cdot -5 - \frac{c}{b} \cdot \frac{c}{b}, a, -c\right)\right)}{b} \]

13. Final simplification 97.4%

    \[\leadsto \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c}{b \cdot b} \cdot \frac{c \cdot c}{b \cdot b}, \mathsf{fma}\left(-5 \cdot \frac{\left(\left(c \cdot a\right) \cdot c\right) \cdot \left(\left(c \cdot a\right) \cdot c\right)}{{b}^{6}} - \frac{c}{b} \cdot \frac{c}{b}, a, -c\right)\right)}{b} \]
14. Add Preprocessing

Alternative 3: 93.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 97.4%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 96.3%

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

10. Applied rewrites 96.4%

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

11. Final simplification 96.4%

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

Alternative 4: 93.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 97.4%

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

7. Applied rewrites 97.4%

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

8. Taylor expanded in b around inf

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

10. Applied rewrites 96.4%

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

11. Final simplification 96.4%

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

Alternative 5: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b \cdot b}\\ \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, t\_0 \cdot \frac{c \cdot c}{b \cdot b}, \mathsf{fma}\left(-a, t\_0, -1\right) \cdot c\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (* b b))))
   (/
    (fma
     (* (* -2.0 a) a)
     (* t_0 (/ (* c c) (* b b)))
     (* (fma (- a) t_0 -1.0) c))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = c / (b * b);
	return fma(((-2.0 * a) * a), (t_0 * ((c * c) / (b * b))), (fma(-a, t_0, -1.0) * c)) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(b * b))
	return Float64(fma(Float64(Float64(-2.0 * a) * a), Float64(t_0 * Float64(Float64(c * c) / Float64(b * b))), Float64(fma(Float64(-a), t_0, -1.0) * c)) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * a), $MachinePrecision] * N[(t$95$0 * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[((-a) * t$95$0 + -1.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 97.4%

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

7. Applied rewrites 97.4%

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

8. Taylor expanded in c around 0

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

10. Applied rewrites 96.3%

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

11. Final simplification 96.3%

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

Alternative 6: 93.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 97.4%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 96.3%

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

10. Applied rewrites 96.3%

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

11. Final simplification 96.3%

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

Alternative 7: 90.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.7%

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

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 29.7

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 29.7

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

4. Applied rewrites 29.7%

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

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. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 93.2

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

7. Applied rewrites 93.2%

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

8. Final simplification 93.2%

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

Alternative 8: 90.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 97.4%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 96.3%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 93.1%

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

Alternative 9: 80.9% 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 29.7%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 83.0

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

5. Applied rewrites 83.0%

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

Alternative 10: 3.2% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 29.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 29.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 30.6

       \[\leadsto \mathsf{fma}\left(-b, \color{blue}{\frac{0.5}{a}}, \frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{2 \cdot a}\right) \]

   13. lift-/.f64 N/A

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

   14. clear-num N/A

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

   15. associate-/r/ N/A

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

   16. lower-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. associate-/r* N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f64 31.1

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

   21. lift-fma.f64 N/A

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

6. Applied rewrites 31.1%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.2

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

9. Applied rewrites 3.2%

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

Reproduce

herbie shell --seed 2024249 
(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)))