Quadratic roots, medium range

Percentage Accurate: 31.2% → 95.4%

Time: 15.4s

Alternatives: 9

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: 31.2% 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, 0.3× 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 33.2%

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

   1. *-commutative 33.2%

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

3. Simplified 33.3%

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

5. Taylor expanded in b around inf 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in a around 0 95.1%

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

   1. associate-*r/ 95.1%

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

10. Applied egg-rr 95.1%

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

    1. associate-*r/ 95.1%

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

    2. mul-1-neg 95.1%

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

    3. unpow2 95.1%

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

    4. unpow2 95.1%

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

    5. times-frac 95.1%

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

    6. sqr-neg 95.1%

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

    7. distribute-frac-neg 95.1%

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

    8. distribute-frac-neg 95.1%

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

    9. unpow2 95.1%

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

    10. distribute-frac-neg 95.1%

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

    11. distribute-frac-neg2 95.1%

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

12. Simplified 95.1%

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

    1. unpow2 95.1%

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

    2. distribute-frac-neg2 95.1%

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

    3. distribute-frac-neg2 95.1%

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

14. Applied egg-rr 95.1%

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

15. Final simplification 95.1%

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

Alternative 2: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-1 + c \cdot \left(c \cdot \left({a}^{3} \cdot \left(-5 \cdot \frac{c}{{b}^{6}} - \frac{2}{a \cdot {b}^{4}}\right)\right) - \frac{a}{{b}^{2}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (+
    -1.0
    (*
     c
     (-
      (*
       c
       (*
        (pow a 3.0)
        (- (* -5.0 (/ c (pow b 6.0))) (/ 2.0 (* a (pow b 4.0))))))
      (/ a (pow b 2.0))))))
  b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((c * (pow(a, 3.0) * ((-5.0 * (c / pow(b, 6.0))) - (2.0 / (a * pow(b, 4.0)))))) - (a / pow(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 * ((-1.0d0) + (c * ((c * ((a ** 3.0d0) * (((-5.0d0) * (c / (b ** 6.0d0))) - (2.0d0 / (a * (b ** 4.0d0)))))) - (a / (b ** 2.0d0)))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((c * (Math.pow(a, 3.0) * ((-5.0 * (c / Math.pow(b, 6.0))) - (2.0 / (a * Math.pow(b, 4.0)))))) - (a / Math.pow(b, 2.0)))))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 + (c * ((c * (math.pow(a, 3.0) * ((-5.0 * (c / math.pow(b, 6.0))) - (2.0 / (a * math.pow(b, 4.0)))))) - (a / math.pow(b, 2.0)))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 + Float64(c * Float64(Float64(c * Float64((a ^ 3.0) * Float64(Float64(-5.0 * Float64(c / (b ^ 6.0))) - Float64(2.0 / Float64(a * (b ^ 4.0)))))) - Float64(a / (b ^ 2.0)))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (-1.0 + (c * ((c * ((a ^ 3.0) * ((-5.0 * (c / (b ^ 6.0))) - (2.0 / (a * (b ^ 4.0)))))) - (a / (b ^ 2.0)))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-1.0 + N[(c * N[(N[(c * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(-5.0 * N[(c / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(a * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 33.2%

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

   1. *-commutative 33.2%

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

3. Simplified 33.3%

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

5. Taylor expanded in b around inf 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in c around 0 95.0%

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

9. Taylor expanded in a around inf 95.0%

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

    1. associate-*r/ 95.0%

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

    2. metadata-eval 95.0%

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

11. Simplified 95.0%

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

12. Final simplification 95.0%

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

Alternative 3: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(c \cdot \left({a}^{3} \cdot \left(-5 \cdot \frac{c}{{b}^{7}} - \frac{2}{a \cdot {b}^{5}}\right)\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (-
     (*
      c
      (* (pow a 3.0) (- (* -5.0 (/ c (pow b 7.0))) (/ 2.0 (* a (pow b 5.0))))))
     (/ a (pow b 3.0))))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((c * ((c * (pow(a, 3.0) * ((-5.0 * (c / pow(b, 7.0))) - (2.0 / (a * pow(b, 5.0)))))) - (a / pow(b, 3.0)))) + (-1.0 / 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 * ((c * ((c * ((a ** 3.0d0) * (((-5.0d0) * (c / (b ** 7.0d0))) - (2.0d0 / (a * (b ** 5.0d0)))))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((c * (Math.pow(a, 3.0) * ((-5.0 * (c / Math.pow(b, 7.0))) - (2.0 / (a * Math.pow(b, 5.0)))))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * ((c * (math.pow(a, 3.0) * ((-5.0 * (c / math.pow(b, 7.0))) - (2.0 / (a * math.pow(b, 5.0)))))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(c * Float64((a ^ 3.0) * Float64(Float64(-5.0 * Float64(c / (b ^ 7.0))) - Float64(2.0 / Float64(a * (b ^ 5.0)))))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((c * ((a ^ 3.0) * ((-5.0 * (c / (b ^ 7.0))) - (2.0 / (a * (b ^ 5.0)))))) - (a / (b ^ 3.0)))) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(c * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(-5.0 * N[(c / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.2%

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

   1. *-commutative 33.2%

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

3. Simplified 33.3%

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

5. Taylor expanded in b around inf 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in c around 0 94.8%

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

9. Taylor expanded in a around inf 94.8%

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

    1. associate-*r/ 94.8%

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

    2. metadata-eval 94.8%

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

11. Simplified 94.8%

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

12. Final simplification 94.8%

    \[\leadsto c \cdot \left(c \cdot \left(c \cdot \left({a}^{3} \cdot \left(-5 \cdot \frac{c}{{b}^{7}} - \frac{2}{a \cdot {b}^{5}}\right)\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right) \]
13. Add Preprocessing

Alternative 4: 93.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

   1. *-commutative 33.2%

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

3. Simplified 33.2%

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

   1. add-cbrt-cube 33.0%

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

   2. pow1/3 33.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 33.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 33.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 33.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 33.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 33.8%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 33.1%

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

8. Simplified 33.1%

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

9. Taylor expanded in b around inf 93.3%

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

11. Simplified 93.3%

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

12. Final simplification 93.3%

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

Alternative 5: 94.0% accurate, 0.4× speedup

Math

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

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 33.2%

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

   1. *-commutative 33.2%

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

3. Simplified 33.3%

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

5. Taylor expanded in b around inf 95.1%

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

7. Simplified 95.1%

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

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. fmm-def 93.3%

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

   2. mul-1-neg 93.3%

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

   3. unsub-neg 93.3%

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

   4. associate-/l* 93.3%

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

   5. unpow2 93.3%

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

   6. unpow2 93.3%

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

   7. times-frac 93.3%

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

   8. sqr-neg 93.3%

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

   9. distribute-frac-neg2 93.3%

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

   10. distribute-frac-neg2 93.3%

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

   11. unpow2 93.3%

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

   12. distribute-frac-neg2 93.3%

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

   13. distribute-neg-frac 93.3%

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

10. Simplified 93.3%

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

    1. *-un-lft-identity 93.3%

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

    2. fmm-undef 93.3%

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

    3. associate-*r* 93.3%

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

    4. fmm-def 93.3%

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

    5. div-inv 93.3%

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

    6. pow-flip 93.3%

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

    7. metadata-eval 93.3%

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

12. Applied egg-rr 93.3%

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

    1. *-lft-identity 93.3%

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

    2. fmm-undef 93.3%

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

    3. associate-*r* 93.3%

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

    4. distribute-frac-neg 93.3%

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

    5. distribute-frac-neg2 93.3%

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

14. Simplified 93.3%

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

Alternative 6: 93.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

   1. *-commutative 33.2%

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

3. Simplified 33.3%

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

5. Taylor expanded in b around inf 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in c around 0 95.0%

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

9. Taylor expanded in a around 0 93.2%

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

10. Final simplification 93.2%

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

Alternative 7: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (* a (+ (* -2.0 (/ (* a c) (pow b 5.0))) (/ -1.0 (pow b 3.0)))))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((c * (a * ((-2.0 * ((a * c) / pow(b, 5.0))) + (-1.0 / pow(b, 3.0))))) + (-1.0 / 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 * ((c * (a * (((-2.0d0) * ((a * c) / (b ** 5.0d0))) + ((-1.0d0) / (b ** 3.0d0))))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * (a * ((-2.0 * ((a * c) / Math.pow(b, 5.0))) + (-1.0 / Math.pow(b, 3.0))))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * (a * ((-2.0 * ((a * c) / math.pow(b, 5.0))) + (-1.0 / math.pow(b, 3.0))))) + (-1.0 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * c) / (b ^ 5.0))) + Float64(-1.0 / (b ^ 3.0))))) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * (a * ((-2.0 * ((a * c) / (b ^ 5.0))) + (-1.0 / (b ^ 3.0))))) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(a * N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.2%

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

   1. *-commutative 33.2%

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

3. Simplified 33.3%

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

5. Taylor expanded in b around inf 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in c around 0 94.8%

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

9. Taylor expanded in a around 0 93.0%

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

10. Final simplification 93.0%

    \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right) \]
11. Add Preprocessing

Alternative 8: 90.9% 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 33.2%

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

   1. *-commutative 33.2%

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

3. Simplified 33.3%

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

5. Taylor expanded in a around 0 89.6%

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

   1. mul-1-neg 89.6%

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

   2. unsub-neg 89.6%

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

   3. mul-1-neg 89.6%

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

   4. distribute-neg-frac2 89.6%

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

   5. associate-/l* 89.6%

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

7. Simplified 89.6%

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

   1. unpow2 89.6%

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

9. Applied egg-rr 89.6%

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

10. Taylor expanded in b around inf 89.6%

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

    1. neg-mul-1 89.6%

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

    2. +-commutative 89.6%

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

    3. unsub-neg 89.6%

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

    4. mul-1-neg 89.6%

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

    5. associate-/l* 89.6%

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

    6. unpow2 89.6%

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

    7. unpow2 89.6%

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

    8. times-frac 89.6%

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

    9. sqr-neg 89.6%

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

    10. unpow2 89.6%

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

    11. distribute-lft-neg-in 89.6%

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

    12. distribute-neg-frac 89.6%

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

12. Simplified 89.6%

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

13. Final simplification 89.6%

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

Alternative 9: 81.4% 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 33.2%

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

   1. *-commutative 33.2%

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

3. Simplified 33.3%

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

5. Taylor expanded in b around inf 79.6%

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

   1. associate-*r/ 79.6%

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

   2. mul-1-neg 79.6%

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

7. Simplified 79.6%

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

8. Final simplification 79.6%

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

Reproduce

herbie shell --seed 2024163 
(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)))