Quadratic roots, medium range

Percentage Accurate: 31.5% → 95.4%

Time: 11.5s

Alternatives: 7

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: 31.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -5, {c}^{4}, \left(\mathsf{fma}\left(-2 \cdot a, c, \left(-b\right) \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/
   (fma
    (* (* a a) -5.0)
    (pow c 4.0)
    (* (* (fma (* -2.0 a) c (* (- b) b)) (* c c)) (* b b)))
   (pow b 7.0))
  a
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	return fma((fma(((a * a) * -5.0), pow(c, 4.0), ((fma((-2.0 * a), c, (-b * b)) * (c * c)) * (b * b))) / pow(b, 7.0)), a, (-c / b));
}

Julia

function code(a, b, c)
	return fma(Float64(fma(Float64(Float64(a * a) * -5.0), (c ^ 4.0), Float64(Float64(fma(Float64(-2.0 * a), c, Float64(Float64(-b) * b)) * Float64(c * c)) * Float64(b * b))) / (b ^ 7.0)), a, Float64(Float64(-c) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -5.0), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * c + N[((-b) * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 94.7%

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

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 94.7%

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

9. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{-5 \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(-2 \cdot \left(a \cdot {c}^{3}\right) + -1 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
10. Step-by-step derivation

11. Applied rewrites 94.7%

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

12. Taylor expanded in c around 0

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left({c}^{2} \cdot \left(-2 \cdot \left(a \cdot c\right) + -1 \cdot {b}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
13. Step-by-step derivation

14. Applied rewrites 94.7%

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

15. Final simplification 94.7%

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

Alternative 2: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-c, -0.5 \cdot \frac{a \cdot a}{{b}^{3}}, \frac{a}{b} \cdot 0.5\right), -0.5 \cdot b\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.5
  (/
   (fma
    c
    (fma (- c) (* -0.5 (/ (* a a) (pow b 3.0))) (* (/ a b) 0.5))
    (* -0.5 b))
   c)))

C

double code(double a, double b, double c) {
	return 0.5 / (fma(c, fma(-c, (-0.5 * ((a * a) / pow(b, 3.0))), ((a / b) * 0.5)), (-0.5 * b)) / c);
}

Julia

function code(a, b, c)
	return Float64(0.5 / Float64(fma(c, fma(Float64(-c), Float64(-0.5 * Float64(Float64(a * a) / (b ^ 3.0))), Float64(Float64(a / b) * 0.5)), Float64(-0.5 * b)) / c))
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(N[(c * N[((-c) * N[(-0.5 * N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 33.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.8

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

4. Applied rewrites 33.8%

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

5. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 93.0%

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

8. Final simplification 93.0%

   \[\leadsto \frac{0.5}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-c, -0.5 \cdot \frac{a \cdot a}{{b}^{3}}, \frac{a}{b} \cdot 0.5\right), -0.5 \cdot b\right)}{c}} \]
9. Add Preprocessing

Alternative 3: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\mathsf{fma}\left(-0.5, \frac{b}{c}, \mathsf{fma}\left(-1, \left(\frac{c}{{b}^{3}} \cdot -0.5\right) \cdot a, \frac{0.5}{b}\right) \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.5
  (fma
   -0.5
   (/ b c)
   (* (fma -1.0 (* (* (/ c (pow b 3.0)) -0.5) a) (/ 0.5 b)) a))))

C

double code(double a, double b, double c) {
	return 0.5 / fma(-0.5, (b / c), (fma(-1.0, (((c / pow(b, 3.0)) * -0.5) * a), (0.5 / b)) * a));
}

Julia

function code(a, b, c)
	return Float64(0.5 / fma(-0.5, Float64(b / c), Float64(fma(-1.0, Float64(Float64(Float64(c / (b ^ 3.0)) * -0.5) * a), Float64(0.5 / b)) * a)))
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(-0.5 * N[(b / c), $MachinePrecision] + N[(N[(-1.0 * N[(N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * a), $MachinePrecision] + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 33.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.8

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

4. Applied rewrites 33.8%

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

5. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 93.0

       \[\leadsto \frac{0.5}{\mathsf{fma}\left(-0.5, \frac{b}{c}, a \cdot \mathsf{fma}\left(-1, a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.5\right), \color{blue}{\frac{0.5}{b}}\right)\right)} \]

7. Applied rewrites 93.0%

   \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{b}{c}, a \cdot \mathsf{fma}\left(-1, a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.5\right), \frac{0.5}{b}\right)\right)}} \]

8. Final simplification 93.0%

   \[\leadsto \frac{0.5}{\mathsf{fma}\left(-0.5, \frac{b}{c}, \mathsf{fma}\left(-1, \left(\frac{c}{{b}^{3}} \cdot -0.5\right) \cdot a, \frac{0.5}{b}\right) \cdot a\right)} \]
9. Add Preprocessing

Alternative 4: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 33.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.8

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

4. Applied rewrites 33.8%

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

5. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 89.5

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

7. Applied rewrites 89.5%

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

8. Final simplification 89.5%

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

Alternative 5: 90.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 33.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.8

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

4. Applied rewrites 33.8%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 89.5

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

7. Applied rewrites 89.5%

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

8. Final simplification 89.5%

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

Alternative 6: 81.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 79.4

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

5. Applied rewrites 79.4%

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

Alternative 7: 3.2% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 33.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 33.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.8

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

4. Applied rewrites 33.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. associate-/r* N/A

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

   17. lift-*.f64 N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 35.4%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.2

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

9. Applied rewrites 3.2%

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

Reproduce

herbie shell --seed 2024270 
(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)))