Quadratic roots, medium range

Percentage Accurate: 31.9% → 95.5%

Time: 12.1s

Alternatives: 5

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{c}{b} \cdot \frac{c}{b}\right) - c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (*
    a
    (-
     (*
      a
      (+
       (* -5.0 (/ (* a (pow c 4.0)) (pow b 6.0)))
       (* -2.0 (/ (pow c 3.0) (pow b 4.0)))))
     (* (/ c b) (/ c b))))
   c)
  b))

C

double code(double a, double b, double c) {
	return ((a * ((a * ((-5.0 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-2.0 * (pow(c, 3.0) / pow(b, 4.0))))) - ((c / b) * (c / b)))) - 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 = ((a * ((a * (((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 6.0d0))) + ((-2.0d0) * ((c ** 3.0d0) / (b ** 4.0d0))))) - ((c / b) * (c / b)))) - c) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((a * ((a * ((-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + (-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 4.0))))) - ((c / b) * (c / b)))) - c) / b;
}

Python

def code(a, b, c):
	return ((a * ((a * ((-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 6.0))) + (-2.0 * (math.pow(c, 3.0) / math.pow(b, 4.0))))) - ((c / b) * (c / b)))) - c) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(Float64(a * Float64(Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 6.0))) + Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 4.0))))) - Float64(Float64(c / b) * Float64(c / b)))) - c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a * ((a * ((-5.0 * ((a * (c ^ 4.0)) / (b ^ 6.0))) + (-2.0 * ((c ^ 3.0) / (b ^ 4.0))))) - ((c / b) * (c / b)))) - c) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(N[(a * N[(N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 29.4%

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

   1. *-commutative 29.4%

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

3. Simplified 29.5%

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

5. Taylor expanded in b around inf 94.5%

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.25 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
6. Step-by-step derivation

7. Simplified 94.5%

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

8. Taylor expanded in a around 0 94.5%

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

   1. associate-*r/ 94.5%

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

10. Applied egg-rr 94.5%

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

    1. mul-1-neg 94.5%

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

    2. distribute-frac-neg 94.5%

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

    3. unpow2 94.5%

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

    4. unpow2 94.5%

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

    5. times-frac 94.5%

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

    6. sqr-neg 94.5%

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

    7. distribute-frac-neg 94.5%

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

    8. distribute-frac-neg 94.5%

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

    9. unpow2 94.5%

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

    10. distribute-frac-neg 94.5%

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

    11. distribute-frac-neg2 94.5%

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

12. Simplified 94.5%

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

    1. unpow2 94.5%

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

    2. distribute-frac-neg2 94.5%

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

    3. distribute-frac-neg2 94.5%

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

14. Applied egg-rr 94.5%

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

15. Final simplification 94.5%

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

Alternative 2: 94.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((a * ((-2.0 * (a * (pow(c, 3.0) / pow(b, 4.0)))) - pow((c / -b), 2.0))) - 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 = ((a * (((-2.0d0) * (a * ((c ** 3.0d0) / (b ** 4.0d0)))) - ((c / -b) ** 2.0d0))) - c) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((a * ((-2.0 * (a * (Math.pow(c, 3.0) / Math.pow(b, 4.0)))) - Math.pow((c / -b), 2.0))) - c) / b;
}

Python

def code(a, b, c):
	return ((a * ((-2.0 * (a * (math.pow(c, 3.0) / math.pow(b, 4.0)))) - math.pow((c / -b), 2.0))) - c) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((a * ((-2.0 * (a * ((c ^ 3.0) / (b ^ 4.0)))) - ((c / -b) ^ 2.0))) - c) / b;
end

Wolfram

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

Derivation

1. Initial program 29.4%

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

   1. *-commutative 29.4%

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

3. Simplified 29.5%

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

5. Taylor expanded in b around inf 94.5%

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.25 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
6. Step-by-step derivation

7. Simplified 94.5%

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

8. Taylor expanded in a around 0 93.3%

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

   1. mul-1-neg 93.3%

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

   2. unsub-neg 93.3%

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

   3. associate-/l* 93.3%

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

   4. unpow2 93.3%

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

   5. unpow2 93.3%

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

   6. times-frac 93.3%

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

   7. sqr-neg 93.3%

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

   8. distribute-frac-neg2 93.3%

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

   9. distribute-frac-neg2 93.3%

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

   10. unpow2 93.3%

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

   11. distribute-frac-neg2 93.3%

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

   12. distribute-neg-frac 93.3%

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

10. Simplified 93.3%

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

11. Final simplification 93.3%

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

Alternative 3: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c + (a * pow((c / -b), 2.0))) / -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 + (a * ((c / -b) ** 2.0d0))) / -b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.4%

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

   1. *-commutative 29.4%

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

3. Simplified 29.5%

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

5. Taylor expanded in b around inf 94.5%

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.25 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
6. Step-by-step derivation

7. Simplified 94.5%

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

8. Taylor expanded in b around inf 90.9%

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

   1. distribute-lft-out 90.9%

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

   2. associate-*r/ 90.9%

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

   3. mul-1-neg 90.9%

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

   4. distribute-neg-frac2 90.9%

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

   5. +-commutative 90.9%

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

   6. associate-/l* 90.9%

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

   7. fma-define 90.9%

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

   8. unpow2 90.9%

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

   9. unpow2 90.9%

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

   10. times-frac 90.9%

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

   11. sqr-neg 90.9%

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

   12. distribute-frac-neg2 90.9%

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

   13. distribute-frac-neg2 90.9%

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

   14. unpow2 90.9%

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

   15. distribute-frac-neg2 90.9%

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

   16. distribute-neg-frac 90.9%

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

10. Simplified 90.9%

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

    1. fma-undefine 90.9%

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

12. Applied egg-rr 90.9%

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

13. Final simplification 90.9%

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

Alternative 4: 90.6% accurate, 7.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((((c / b) * (c / b)) + (c / a)) * -a) / 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) * (c / b)) + (c / a)) * -a) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.4%

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

   1. *-commutative 29.4%

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

3. Simplified 29.5%

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

5. Taylor expanded in b around inf 94.5%

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.25 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
6. Step-by-step derivation

7. Simplified 94.5%

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

8. Taylor expanded in b around inf 90.9%

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

   1. distribute-lft-out 90.9%

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

   2. associate-*r/ 90.9%

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

   3. mul-1-neg 90.9%

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

   4. distribute-neg-frac2 90.9%

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

   5. +-commutative 90.9%

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

   6. associate-/l* 90.9%

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

   7. fma-define 90.9%

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

   8. unpow2 90.9%

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

   9. unpow2 90.9%

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

   10. times-frac 90.9%

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

   11. sqr-neg 90.9%

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

   12. distribute-frac-neg2 90.9%

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

   13. distribute-frac-neg2 90.9%

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

   14. unpow2 90.9%

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

   15. distribute-frac-neg2 90.9%

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

   16. distribute-neg-frac 90.9%

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

10. Simplified 90.9%

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

11. Taylor expanded in a around inf 90.7%

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

    1. unpow2 90.7%

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

    2. unpow2 90.7%

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

    3. times-frac 90.7%

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

    4. sqr-neg 90.7%

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

    5. distribute-frac-neg 90.7%

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

    6. distribute-frac-neg 90.7%

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

    7. unpow2 90.7%

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

    8. distribute-frac-neg 90.7%

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

    9. distribute-frac-neg2 90.7%

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

13. Simplified 90.7%

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

    1. unpow2 94.5%

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

    2. distribute-frac-neg2 94.5%

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

    3. distribute-frac-neg2 94.5%

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

15. Applied egg-rr 90.7%

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

16. Final simplification 90.7%

    \[\leadsto \frac{\left(\frac{c}{b} \cdot \frac{c}{b} + \frac{c}{a}\right) \cdot \left(-a\right)}{b} \]
17. Add Preprocessing

Alternative 5: 81.0% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.4%

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

   1. *-commutative 29.4%

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

3. Simplified 29.5%

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

5. Taylor expanded in b around inf 82.4%

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

   1. associate-*r/ 82.4%

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

   2. mul-1-neg 82.4%

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

7. Simplified 82.4%

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

8. Final simplification 82.4%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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)))