Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.7%

Time: 9.6s

Alternatives: 11

Speedup: 3.6×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{-c}{b \cdot b}, \frac{c}{b}, \left(\left({c}^{3} \cdot \mathsf{fma}\left(c \cdot a, -5, -2 \cdot \left(b \cdot b\right)\right)\right) \cdot {b}^{-7}\right) \cdot a\right), a, \frac{-c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (fma
   (/ (- c) (* b b))
   (/ c b)
   (* (* (* (pow c 3.0) (fma (* c a) -5.0 (* -2.0 (* b b)))) (pow b -7.0)) a))
  a
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	return fma(fma((-c / (b * b)), (c / b), (((pow(c, 3.0) * fma((c * a), -5.0, (-2.0 * (b * b)))) * pow(b, -7.0)) * a)), a, (-c / b));
}

Julia

function code(a, b, c)
	return fma(fma(Float64(Float64(-c) / Float64(b * b)), Float64(c / b), Float64(Float64(Float64((c ^ 3.0) * fma(Float64(c * a), -5.0, Float64(-2.0 * Float64(b * b)))) * (b ^ -7.0)) * a)), a, Float64(Float64(-c) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / b), $MachinePrecision] + N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(c * a), $MachinePrecision] * -5.0 + N[(-2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[b, -7.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 97.1%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 97.1%

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

13. Applied rewrites 97.1%

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

14. Final simplification 97.1%

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

Alternative 2: 97.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (/ (* (* (* (pow c 3.0) a) a) -2.0) (pow b 4.0))
   (fma (/ c b) (/ (* c a) b) c))
  b))

C

double code(double a, double b, double c) {
	return (((((pow(c, 3.0) * a) * a) * -2.0) / pow(b, 4.0)) - fma((c / b), ((c * a) / b), c)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64((c ^ 3.0) * a) * a) * -2.0) / (b ^ 4.0)) - fma(Float64(c / b), Float64(Float64(c * a) / b), c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 16.0%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 96.4%

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

Alternative 3: 97.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-c\right) - \mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (- c)
   (fma 2.0 (/ (* (* a a) (pow c 3.0)) (pow b 4.0)) (/ (* (* c c) a) (* b b))))
  b))

C

double code(double a, double b, double c) {
	return (-c - fma(2.0, (((a * a) * pow(c, 3.0)) / pow(b, 4.0)), (((c * c) * a) / (b * b)))) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-c) - fma(2.0, Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 4.0)), Float64(Float64(Float64(c * c) * a) / Float64(b * b)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[((-c) - N[(2.0 * N[(N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 16.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 97.1%

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

9. Taylor expanded in b around -inf

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

11. Applied rewrites 96.4%

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

12. Final simplification 96.4%

    \[\leadsto \frac{\left(-c\right) - \mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]
13. Add Preprocessing

Alternative 4: 97.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot a}{b \cdot b}, \left(-c\right) \cdot c\right)}{{b}^{3}}, a, \frac{-c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/ (fma -2.0 (/ (* (pow c 3.0) a) (* b b)) (* (- c) c)) (pow b 3.0))
  a
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	return fma((fma(-2.0, ((pow(c, 3.0) * a) / (b * b)), (-c * c)) / pow(b, 3.0)), a, (-c / b));
}

Julia

function code(a, b, c)
	return fma(Float64(fma(-2.0, Float64(Float64((c ^ 3.0) * a) / Float64(b * b)), Float64(Float64(-c) * c)) / (b ^ 3.0)), a, Float64(Float64(-c) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[((-c) * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 97.1%

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

9. Taylor expanded in b around inf

   \[\leadsto \mathsf{fma}\left(\frac{-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + -1 \cdot {c}^{2}}{{b}^{3}}, a, \frac{-c}{b}\right) \]
10. Step-by-step derivation

11. Applied rewrites 96.4%

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

12. Final simplification 96.4%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot a}{b \cdot b}, \left(-c\right) \cdot c\right)}{{b}^{3}}, a, \frac{-c}{b}\right) \]
13. Add Preprocessing

Alternative 5: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -2, c, \left(\left(-b\right) \cdot b\right) \cdot a\right)}{{b}^{5}}, c, \frac{-1}{b}\right) \cdot c \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (fma (/ (fma (* (* a a) -2.0) c (* (* (- b) b) a)) (pow b 5.0)) c (/ -1.0 b))
  c))

C

double code(double a, double b, double c) {
	return fma((fma(((a * a) * -2.0), c, ((-b * b) * a)) / pow(b, 5.0)), c, (-1.0 / b)) * c;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(fma(Float64(Float64(a * a) * -2.0), c, Float64(Float64(Float64(-b) * b) * a)) / (b ^ 5.0)), c, Float64(-1.0 / b)) * c)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] * c + N[(N[((-b) * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 16.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 96.0%

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

9. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{-2 \cdot \left({a}^{2} \cdot c\right) + -1 \cdot \left(a \cdot {b}^{2}\right)}{{b}^{5}}, c, \frac{-1}{b}\right) \cdot c \]
10. Step-by-step derivation

11. Applied rewrites 96.0%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2 \cdot \left(a \cdot a\right), c, \left(-a\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}}, c, \frac{-1}{b}\right) \cdot c \]

12. Final simplification 96.0%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -2, c, \left(\left(-b\right) \cdot b\right) \cdot a\right)}{{b}^{5}}, c, \frac{-1}{b}\right) \cdot c \]
13. Add Preprocessing

Alternative 6: 96.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(N[((-b) * b), $MachinePrecision] * N[(a * c + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 16.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 96.0%

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

9. Taylor expanded in b around 0

   \[\leadsto \frac{-2 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(-1 \cdot \left(a \cdot c\right) + -1 \cdot {b}^{2}\right)}{{b}^{5}} \cdot c \]
10. Step-by-step derivation

11. Applied rewrites 95.6%

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

12. Final simplification 95.6%

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

Alternative 7: 95.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (- (fma a (/ (* c c) (pow b 3.0)) (/ c b))))

C

double code(double a, double b, double c) {
	return -fma(a, ((c * c) / pow(b, 3.0)), (c / b));
}

Julia

function code(a, b, c)
	return Float64(-fma(a, Float64(Float64(c * c) / (b ^ 3.0)), Float64(c / b)))
end

Wolfram

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

Derivation

1. Initial program 16.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-out N/A

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

   5. lower-neg.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-/.f64 95.2

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

8. Applied rewrites 95.2%

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

Alternative 8: 95.4% 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 16.0%

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

   1. lift--.f64 N/A

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

   2. 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 16.0

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

4. Applied rewrites 16.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sub-neg N/A

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

   10. flip-- N/A

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

6. Applied rewrites 16.2%

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

7. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
8. 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 95.2

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

9. Applied rewrites 95.2%

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

10. Final simplification 95.2%

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

Alternative 9: 95.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 96.0%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 94.8%

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

Alternative 10: 90.5% 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 16.0%

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

3. Taylor expanded in 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 91.4

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

5. Applied rewrites 91.4%

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

Alternative 11: 3.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 16.0%

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

   1. lift--.f64 N/A

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

   2. 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 16.0

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

4. Applied rewrites 16.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sub-neg N/A

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

   10. flip-- N/A

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

6. Applied rewrites 16.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

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

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

   8. lower-fma.f64 N/A

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

8. Applied rewrites 16.3%

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

9. Taylor expanded in c around 0

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

    1. associate-*r/ N/A

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

    2. distribute-rgt-out N/A

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

    3. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left(b \cdot \color{blue}{0}\right)}{a} \]

    4. mul0-rgt N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{0}}{a} \]

    5. metadata-eval N/A

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

    6. lower-/.f64 3.3

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

11. Applied rewrites 3.3%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024248 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))