Quadratic roots, medium range

Percentage Accurate: 31.7% → 95.3%

Time: 15.0s

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: 31.7% 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: 95.3% 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 29.3%

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

   1. *-commutative 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in b around inf 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in a around 0 98.0%

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

   1. associate-*r/ 98.0%

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

10. Applied egg-rr 98.0%

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

    1. associate-*r/ 98.0%

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

    2. mul-1-neg 98.0%

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

    3. unpow2 98.0%

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

    4. unpow2 98.0%

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

    5. times-frac 98.0%

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

    6. sqr-neg 98.0%

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

    7. distribute-frac-neg 98.0%

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

    8. distribute-frac-neg 98.0%

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

    9. unpow1 98.0%

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

    10. pow-plus 98.0%

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

    11. distribute-frac-neg 98.0%

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

    12. distribute-neg-frac2 98.0%

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

    13. metadata-eval 98.0%

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

12. Simplified 98.0%

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

    1. unpow2 98.0%

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

14. Applied egg-rr 98.0%

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

15. Final simplification 98.0%

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

Alternative 2: 93.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.3%

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

   1. *-commutative 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in b around inf 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in a around 0 96.7%

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

   1. fmm-def 96.7%

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

   2. mul-1-neg 96.7%

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

   3. unsub-neg 96.7%

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

   4. associate-/l* 96.7%

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

   5. unpow2 96.7%

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

   6. unpow2 96.7%

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

   7. times-frac 96.7%

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

   8. sqr-neg 96.7%

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

   9. distribute-frac-neg2 96.7%

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

   10. distribute-frac-neg2 96.7%

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

   11. unpow2 96.7%

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

   12. distribute-frac-neg2 96.7%

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

   13. distribute-neg-frac 96.7%

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

10. Simplified 96.7%

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

    1. *-un-lft-identity 96.7%

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

    2. associate-*r/ 96.7%

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

    3. clear-num 96.7%

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

    4. inv-pow 96.7%

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

    5. pow-pow 96.7%

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

    6. distribute-frac-neg2 96.7%

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

    7. metadata-eval 96.7%

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

12. Applied egg-rr 96.7%

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

    1. *-lft-identity 96.7%

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

    2. fmm-undef 96.7%

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

    3. *-commutative 96.7%

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

    4. associate-/l* 96.7%

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

    5. associate-*l* 96.7%

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

    6. distribute-neg-frac2 96.7%

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

14. Simplified 96.7%

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

15. Final simplification 96.7%

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

Alternative 3: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-1 + c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{4}} - \frac{a}{b \cdot b}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (+ -1.0 (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 4.0))) (/ a (* b b))))))
  b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 4.0))) - (a / (b * b)))))) / 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 * ((-1.0d0) + (c * (((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 4.0d0))) - (a / (b * b)))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((-2.0 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 4.0))) - (a / (b * b)))))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 + (c * ((-2.0 * ((c * math.pow(a, 2.0)) / math.pow(b, 4.0))) - (a / (b * b)))))) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (-1.0 + (c * ((-2.0 * ((c * (a ^ 2.0)) / (b ^ 4.0))) - (a / (b * b)))))) / b;
end

Wolfram

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

Derivation

1. Initial program 29.3%

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

   1. *-commutative 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in b around inf 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in c around 0 96.6%

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

   1. unpow2 96.6%

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

10. Applied egg-rr 96.6%

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

11. Final simplification 96.6%

    \[\leadsto \frac{c \cdot \left(-1 + c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{4}} - \frac{a}{b \cdot b}\right)\right)}{b} \]
12. Add Preprocessing

Alternative 4: 90.5% accurate, 1.0× speedup

Math

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

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(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / 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 29.3%

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

   1. *-commutative 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in b around inf 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in b around inf 93.5%

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

   1. neg-mul-1 93.5%

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

   2. +-commutative 93.5%

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

   3. sub-neg 93.5%

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

   4. mul-1-neg 93.5%

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

   5. associate-/l* 93.5%

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

   6. unpow2 93.5%

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

   7. unpow2 93.5%

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

   8. times-frac 93.5%

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

   9. sqr-neg 93.5%

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

   10. distribute-frac-neg2 93.5%

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

   11. distribute-frac-neg2 93.5%

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

   12. unpow2 93.5%

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

   13. distribute-frac-neg2 93.5%

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

   14. distribute-neg-frac 93.5%

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

10. Simplified 93.5%

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

11. Final simplification 93.5%

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

Alternative 5: 81.1% 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 29.3%

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

   1. *-commutative 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in b around inf 83.1%

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

   1. associate-*r/ 83.1%

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

   2. mul-1-neg 83.1%

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

7. Simplified 83.1%

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

8. Final simplification 83.1%

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

Reproduce

herbie shell --seed 2024150 
(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)))