Quadratic roots, medium range

Percentage Accurate: 31.6% → 95.5%

Time: 12.3s

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: 31.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 95.8%

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

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 95.8%

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

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 95.8%

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

12. Final simplification 95.8%

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

Alternative 2: 95.5% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (fma
   (/ (- c) (* b b))
   (/ c b)
   (* (* (* (fma (* c a) -5.0 (* (* b b) -2.0)) (pow c 3.0)) (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), (((fma((c * a), -5.0, ((b * b) * -2.0)) * pow(c, 3.0)) * 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(fma(Float64(c * a), -5.0, Float64(Float64(b * b) * -2.0)) * (c ^ 3.0)) * (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[(N[(c * a), $MachinePrecision] * -5.0 + N[(N[(b * b), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[b, -7.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 95.8%

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

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 95.8%

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

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 95.8%

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

13. Applied rewrites 95.8%

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

14. Final simplification 95.8%

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

Alternative 3: 93.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 94.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 94.1%

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

9. Final simplification 94.1%

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

Alternative 4: 93.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot c\right) \cdot a\\ \frac{\frac{\mathsf{fma}\left(\frac{-2}{b}, \frac{\left(\left(t\_0 \cdot 3\right) \cdot c\right) \cdot a}{b}, -3 \cdot t\_0\right)}{b \cdot b} - c}{{b}^{3}} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) a)))
   (*
    (/
     (-
      (/ (fma (/ -2.0 b) (/ (* (* (* t_0 3.0) c) a) b) (* -3.0 t_0)) (* b b))
      c)
     (pow b 3.0))
    (fma
     (* -4.0 c)
     a
     (fma b b (* (- b (sqrt (fma (* -4.0 c) a (* b b)))) b))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * a;
	return (((fma((-2.0 / b), ((((t_0 * 3.0) * c) * a) / b), (-3.0 * t_0)) / (b * b)) - c) / pow(b, 3.0)) * fma((-4.0 * c), a, fma(b, b, ((b - sqrt(fma((-4.0 * c), a, (b * b)))) * b)));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * a)
	return Float64(Float64(Float64(Float64(fma(Float64(-2.0 / b), Float64(Float64(Float64(Float64(t_0 * 3.0) * c) * a) / b), Float64(-3.0 * t_0)) / Float64(b * b)) - c) / (b ^ 3.0)) * fma(Float64(-4.0 * c), a, fma(b, b, Float64(Float64(b - sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))) * b))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(-2.0 / b), $MachinePrecision] * N[(N[(N[(N[(t$95$0 * 3.0), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] / b), $MachinePrecision] + N[(-3.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(N[(b - N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 31.6%

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

4. Applied rewrites 31.6%

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

5. Taylor expanded in b around inf

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

7. Applied rewrites 93.5%

   \[\leadsto \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(-2 \cdot \mathsf{fma}\left(c \cdot c, a, \left(2 \cdot \left(a \cdot c\right)\right) \cdot c\right)\right) \cdot \left(\left(a \cdot -2\right) \cdot c\right)}{{b}^{4}}, -0.5, \frac{-2 \cdot \mathsf{fma}\left(c \cdot c, a, \left(2 \cdot \left(a \cdot c\right)\right) \cdot c\right)}{b \cdot b} \cdot 0.5\right) - c}{{b}^{3}}} \]

8. Taylor expanded in b around inf

   \[\leadsto \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot \frac{\frac{-2 \cdot \frac{a \cdot \left(c \cdot \left(2 \cdot \left(a \cdot {c}^{2}\right) + a \cdot {c}^{2}\right)\right)}{{b}^{2}} + -1 \cdot \left(2 \cdot \left(a \cdot {c}^{2}\right) + a \cdot {c}^{2}\right)}{{b}^{2}} - c}{{b}^{3}} \]
9. Step-by-step derivation

10. Applied rewrites 93.5%

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

11. Final simplification 93.5%

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

Alternative 5: 93.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

4. Applied rewrites 31.6%

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

5. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 90.4

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

7. Applied rewrites 90.4%

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

8. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r* N/A

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

   9. associate-*r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 93.5

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

10. Applied rewrites 93.5%

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

11. Final simplification 93.5%

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

Alternative 6: 93.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

4. Applied rewrites 31.6%

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

5. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 90.4

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

7. Applied rewrites 90.4%

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

8. Taylor expanded in c around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 93.5

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

10. Applied rewrites 93.5%

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

11. Final simplification 93.5%

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

Alternative 7: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* 2.0 a)) -5e-10)
   (* (- (sqrt (fma (* -4.0 c) a (* b b))) b) (/ 0.5 a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a)) <= -5e-10) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(2.0 * a)) <= -5e-10)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -5e-10], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.00000000000000031e-10

   1. Initial program 68.1%

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

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 68.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 68.1

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

   4. Applied rewrites 68.2%

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

   if -5.00000000000000031e-10 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 11.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 95.2

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

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. 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-/l* 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. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. times-frac N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 90.9

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

5. Applied rewrites 90.9%

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

6. Final simplification 90.9%

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

Alternative 9: 81.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 80.9

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

5. Applied rewrites 80.9%

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

Alternative 10: 3.2% 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 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 31.3%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]
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. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. inv-pow N/A

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

   7. pow-prod-down N/A

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

   8. inv-pow N/A

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

   9. inv-pow N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 33.1%

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

7. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{-2 \cdot b + 2 \cdot b}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. distribute-rgt-out N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(b \cdot \color{blue}{0}\right)}{a} \]

   4. mul0-rgt N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{0}}{a} \]

   5. metadata-eval N/A

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

   6. lower-/.f64 3.2

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

9. Applied rewrites 3.2%

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

Reproduce

herbie shell --seed 2024268 
(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)))