Quadratic roots, wide range

Percentage Accurate: 18.0% → 97.7%

Time: 14.6s

Alternatives: 9

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 18.0% 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: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ a \cdot \left(\frac{\frac{a}{\frac{b}{a \cdot -0.25}}}{\frac{t\_0 \cdot \frac{t\_0}{c \cdot c}}{\left(c \cdot c\right) \cdot 20}} + \frac{\left(c \cdot c\right) \cdot \left(-1 + \frac{a \cdot -2}{\frac{b}{\frac{c}{b}}}\right)}{t\_0}\right) - \frac{c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (-
    (*
     a
     (+
      (/ (/ a (/ b (* a -0.25))) (/ (* t_0 (/ t_0 (* c c))) (* (* c c) 20.0)))
      (/ (* (* c c) (+ -1.0 (/ (* a -2.0) (/ b (/ c b))))) t_0)))
    (/ c b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (a * (((a / (b / (a * -0.25))) / ((t_0 * (t_0 / (c * c))) / ((c * c) * 20.0))) + (((c * c) * (-1.0 + ((a * -2.0) / (b / (c / b))))) / t_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
    real(8) :: t_0
    t_0 = b * (b * b)
    code = (a * (((a / (b / (a * (-0.25d0)))) / ((t_0 * (t_0 / (c * c))) / ((c * c) * 20.0d0))) + (((c * c) * ((-1.0d0) + ((a * (-2.0d0)) / (b / (c / b))))) / t_0))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (a * (((a / (b / (a * -0.25))) / ((t_0 * (t_0 / (c * c))) / ((c * c) * 20.0))) + (((c * c) * (-1.0 + ((a * -2.0) / (b / (c / b))))) / t_0))) - (c / b);
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return (a * (((a / (b / (a * -0.25))) / ((t_0 * (t_0 / (c * c))) / ((c * c) * 20.0))) + (((c * c) * (-1.0 + ((a * -2.0) / (b / (c / b))))) / t_0))) - (c / b)

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(a * Float64(Float64(Float64(a / Float64(b / Float64(a * -0.25))) / Float64(Float64(t_0 * Float64(t_0 / Float64(c * c))) / Float64(Float64(c * c) * 20.0))) + Float64(Float64(Float64(c * c) * Float64(-1.0 + Float64(Float64(a * -2.0) / Float64(b / Float64(c / b))))) / t_0))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = (a * (((a / (b / (a * -0.25))) / ((t_0 * (t_0 / (c * c))) / ((c * c) * 20.0))) + (((c * c) * (-1.0 + ((a * -2.0) / (b / (c / b))))) / t_0))) - (c / b);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(N[(N[(a / N[(b / N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[(t$95$0 / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(-1.0 + N[(N[(a * -2.0), $MachinePrecision] / N[(b / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in a around 0

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

5. Simplified 96.7%

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

7. Applied egg-rr 96.7%

   \[\leadsto a \cdot \color{blue}{\left(\frac{a \cdot \left(c \cdot \left(c \cdot \left(c \cdot -2\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(\frac{a}{\frac{\frac{b}{-0.25}}{\frac{a}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)}}}} + \frac{\frac{c}{\frac{b}{\frac{c}{b}}}}{0 - b}\right)\right)} - \frac{c}{b} \]

9. Applied egg-rr 96.7%

   \[\leadsto \color{blue}{\left(\frac{-2 \cdot \left(c \cdot \frac{c}{\frac{b \cdot \frac{b}{c}}{a}}\right) - c \cdot c}{b \cdot \left(b \cdot b\right)} + \frac{a}{\frac{\frac{\frac{b}{-0.25}}{a}}{\left(\left(c \cdot c\right) \cdot 20\right) \cdot \frac{c \cdot c}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}}\right) \cdot a} - \frac{c}{b} \]

11. Applied egg-rr 96.7%

    \[\leadsto \color{blue}{\left(\frac{\frac{a}{\frac{b}{a \cdot -0.25}}}{\frac{\left(b \cdot \left(b \cdot b\right)\right) \cdot \frac{b \cdot \left(b \cdot b\right)}{c \cdot c}}{\left(c \cdot c\right) \cdot 20}} + \frac{\left(c \cdot c\right) \cdot \left(-1 + \frac{-2 \cdot a}{\frac{b}{\frac{c}{b}}}\right)}{b \cdot \left(b \cdot b\right)}\right)} \cdot a - \frac{c}{b} \]

12. Final simplification 96.7%

    \[\leadsto a \cdot \left(\frac{\frac{a}{\frac{b}{a \cdot -0.25}}}{\frac{\left(b \cdot \left(b \cdot b\right)\right) \cdot \frac{b \cdot \left(b \cdot b\right)}{c \cdot c}}{\left(c \cdot c\right) \cdot 20}} + \frac{\left(c \cdot c\right) \cdot \left(-1 + \frac{a \cdot -2}{\frac{b}{\frac{c}{b}}}\right)}{b \cdot \left(b \cdot b\right)}\right) - \frac{c}{b} \]
13. Add Preprocessing

Alternative 2: 96.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in b around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), \color{blue}{b}\right) \]

5. Simplified 95.8%

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

6. Final simplification 95.8%

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

Alternative 3: 96.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, c):
	return (a * ((((-2.0 * (a * (c * (c * c)))) / (b * b)) - (c * c)) / (b * (b * b)))) - (c / b)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in a around 0

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

5. Simplified 96.7%

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

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} - {c}^{2}}{{b}^{3}}\right)}\right), \mathsf{/.f64}\left(c, b\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} - {c}^{2}\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \left({c}^{3}\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \left(c \cdot \left(c \cdot c\right)\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \left(c \cdot {c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \left(b \cdot b\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(c \cdot c\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   16. cube-mult N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \left(b \cdot \left(b \cdot b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \left(b \cdot {b}^{2}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, \left({b}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   20. *-lowering-*.f64 95.8%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

8. Simplified 95.8%

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

Alternative 4: 96.9% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in a around 0

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

5. Simplified 96.7%

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

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} - {c}^{2}}{{b}^{3}}\right)}\right), \mathsf{/.f64}\left(c, b\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} - {c}^{2}\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \left({c}^{3}\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \left(c \cdot \left(c \cdot c\right)\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \left(c \cdot {c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \left({b}^{2}\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \left(b \cdot b\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left({c}^{2}\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(c \cdot c\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   16. cube-mult N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \left(b \cdot \left(b \cdot b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \left(b \cdot {b}^{2}\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, \left({b}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

   20. *-lowering-*.f64 95.8%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

8. Simplified 95.8%

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

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\color{blue}{\left({c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)\right)}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({c}^{2}\right), \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c \cdot c\right), \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -1\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(-1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(-1, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    9. associate-/l* N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \left(a \cdot \frac{c}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \left(\frac{c}{{b}^{2}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \left({b}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \left(b \cdot b\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

    13. *-lowering-*.f64 95.8%

        \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(c, b\right)\right) \]

11. Simplified 95.8%

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

Alternative 5: 96.6% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in c around 0

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

5. Simplified 96.4%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(c \cdot \left(\frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}} + \frac{\left(-0.25 \cdot c\right) \cdot \frac{{a}^{4} \cdot 20}{{b}^{6}}}{a \cdot b}\right) - \frac{a}{b \cdot \left(b \cdot b\right)}\right) + \frac{-1}{b}\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right) - 1}{b}\right)}\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\left(\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right) - 1\right), \color{blue}{b}\right)\right) \]

8. Simplified 95.4%

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

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(a \cdot \left(-2 \cdot \frac{a \cdot {c}^{2}}{{b}^{4}} - \frac{c}{{b}^{2}}\right)\right)}, -1\right), b\right)\right) \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(-2 \cdot \frac{a \cdot {c}^{2}}{{b}^{4}} + \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right)\right), -1\right), b\right)\right) \]

    2. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(-2 \cdot \frac{a \cdot {c}^{2}}{{b}^{4}}\right) + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(\frac{a \cdot {c}^{2}}{{b}^{4}} \cdot -2\right) + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    4. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(\left(a \cdot \frac{{c}^{2}}{{b}^{4}}\right) \cdot -2\right) + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(a \cdot \left(\frac{{c}^{2}}{{b}^{4}} \cdot -2\right)\right) + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{4}}\right)\right) + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot a\right) \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{4}}\right) + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a}^{2} \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{4}}\right) + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a}^{2} \cdot \left(\frac{{c}^{2}}{{b}^{4}} \cdot -2\right) + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    10. associate-*r* N/A

        \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left({a}^{2} \cdot \frac{{c}^{2}}{{b}^{4}}\right) \cdot -2 + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    11. associate-/l* N/A

        \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} \cdot -2 + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + a \cdot \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    13. distribute-rgt-neg-in N/A

        \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + \left(\mathsf{neg}\left(a \cdot \frac{c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

    14. associate-/l* N/A

        \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + \left(\mathsf{neg}\left(\frac{a \cdot c}{{b}^{2}}\right)\right)\right), -1\right), b\right)\right) \]

11. Simplified 95.4%

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

12. Final simplification 95.4%

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

Alternative 6: 95.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{c + a \cdot \left(c \cdot \frac{\frac{c}{b}}{b}\right)}{0 - b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ c (* a (* c (/ (/ c b) b)))) (- 0.0 b)))

C

double code(double a, double b, double c) {
	return (c + (a * (c * ((c / b) / b)))) / (0.0 - 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 * ((c / b) / b)))) / (0.0d0 - b)
end function

Java

public static double code(double a, double b, double c) {
	return (c + (a * (c * ((c / b) / b)))) / (0.0 - b);
}

Python

def code(a, b, c):
	return (c + (a * (c * ((c / b) / b)))) / (0.0 - b)

Julia

function code(a, b, c)
	return Float64(Float64(c + Float64(a * Float64(c * Float64(Float64(c / b) / b)))) / Float64(0.0 - b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c + (a * (c * ((c / b) / b)))) / (0.0 - b);
end

Wolfram

code[a_, b_, c_] := N[(N[(c + N[(a * N[(c * N[(N[(c / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in b around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right), \color{blue}{b}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -1 \cdot c\right), b\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(\mathsf{neg}\left(c\right)\right)\right), b\right) \]

   4. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} - c\right), b\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right), c\right), b\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), c\right), b\right) \]

   7. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - \frac{a \cdot {c}^{2}}{{b}^{2}}\right), c\right), b\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), c\right), b\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right), c\right), b\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \left(\frac{{c}^{2}}{{b}^{2}}\right)\right)\right), c\right), b\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \left(\frac{c \cdot c}{{b}^{2}}\right)\right)\right), c\right), b\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \left(c \cdot \frac{c}{{b}^{2}}\right)\right)\right), c\right), b\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\frac{c}{{b}^{2}}\right)\right)\right)\right), c\right), b\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\frac{c}{b \cdot b}\right)\right)\right)\right), c\right), b\right) \]

   15. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\frac{\frac{c}{b}}{b}\right)\right)\right)\right), c\right), b\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\left(\frac{c}{b}\right), b\right)\right)\right)\right), c\right), b\right) \]

   17. /-lowering-/.f64 93.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, b\right), b\right)\right)\right)\right), c\right), b\right) \]

5. Simplified 93.7%

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

6. Final simplification 93.7%

   \[\leadsto \frac{c + a \cdot \left(c \cdot \frac{\frac{c}{b}}{b}\right)}{0 - b} \]
7. Add Preprocessing

Alternative 7: 95.3% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (a b c) :precision binary64 (/ (* c (- -1.0 (/ (* a c) (* b b)))) b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 - ((a * c) / (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) - ((a * c) / (b * b)))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 - ((a * c) / (b * b)))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 - ((a * c) / (b * b)))) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 - Float64(Float64(a * c) / Float64(b * b)))) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in b around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \left(a \cdot c\right) + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}}{b}\right)}, \mathsf{*.f64}\left(2, a\right)\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right) + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right), b\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

   2. distribute-lft-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(a \cdot c\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\left(\left(a \cdot a\right) \cdot {c}^{2}\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\left(a \cdot \left(a \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   14. *-lowering-*.f64 93.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

5. Simplified 93.3%

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

   1. associate-/l* N/A

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

   2. times-frac N/A

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

   3. metadata-eval N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\frac{a \cdot c + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot b}}{b}}{a}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{a \cdot c + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot b}}{b}\right), \color{blue}{a}\right)\right) \]

7. Applied egg-rr 93.3%

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

   1. associate-/l/ N/A

      \[\leadsto \mathsf{*.f64}\left(-1, \left(\frac{a \cdot \left(c + \frac{c \cdot \left(a \cdot c\right)}{b \cdot b}\right)}{\color{blue}{a \cdot b}}\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(-1, \left(\frac{\frac{a \cdot \left(c + \frac{c \cdot \left(a \cdot c\right)}{b \cdot b}\right)}{a}}{\color{blue}{b}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{a \cdot \left(c + \frac{c \cdot \left(a \cdot c\right)}{b \cdot b}\right)}{a}\right), \color{blue}{b}\right)\right) \]

9. Applied egg-rr 93.6%

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

10. Taylor expanded in c around 0

    \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\color{blue}{\left(c \cdot \left(1 + \frac{a \cdot c}{{b}^{2}}\right)\right)}, b\right)\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(1 + \frac{a \cdot c}{{b}^{2}}\right)\right), b\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(1, \left(\frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right)\right)\right), b\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(c \cdot a\right), \left({b}^{2}\right)\right)\right)\right), b\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), \left({b}^{2}\right)\right)\right)\right), b\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), \left(b \cdot b\right)\right)\right)\right), b\right)\right) \]

    7. *-lowering-*.f64 93.7%

       \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right)\right) \]

12. Simplified 93.7%

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

13. Final simplification 93.7%

    \[\leadsto \frac{c \cdot \left(-1 - \frac{a \cdot c}{b \cdot b}\right)}{b} \]
14. Add Preprocessing

Alternative 8: 94.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-1 - a \cdot \frac{c}{b \cdot b}}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* c (/ (- -1.0 (* a (/ c (* b b)))) b)))

C

double code(double a, double b, double c) {
	return c * ((-1.0 - (a * (c / (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) - (a * (c / (b * b)))) / b)
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 - (a * (c / (b * b)))) / b);
}

Python

def code(a, b, c):
	return c * ((-1.0 - (a * (c / (b * b)))) / b)

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 - Float64(a * Float64(c / Float64(b * b)))) / b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in b around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \left(a \cdot c\right) + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}}{b}\right)}, \mathsf{*.f64}\left(2, a\right)\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right) + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right), b\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

   2. distribute-lft-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(a \cdot c\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\left(\left(a \cdot a\right) \cdot {c}^{2}\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\left(a \cdot \left(a \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   14. *-lowering-*.f64 93.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right) \]

5. Simplified 93.3%

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

   1. associate-/l* N/A

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

   2. times-frac N/A

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

   3. metadata-eval N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\frac{a \cdot c + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot b}}{b}}{a}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{a \cdot c + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot b}}{b}\right), \color{blue}{a}\right)\right) \]

7. Applied egg-rr 93.3%

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

8. Taylor expanded in a around 0

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

   1. distribute-lft-out N/A

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

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{neg}\left(\left(\frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{c}{b}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\left(\frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + \frac{c}{b}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{neg}\left(\left(\frac{\left(a \cdot c\right) \cdot c}{{b}^{3}} + \frac{c}{b}\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{neg}\left(\left(\frac{a \cdot c}{{b}^{3}} \cdot c + \frac{c}{b}\right)\right) \]

   7. *-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\left(\frac{a \cdot c}{{b}^{3}} \cdot c + \frac{1 \cdot c}{b}\right)\right) \]

   8. associate-*l/ N/A

      \[\leadsto \mathsf{neg}\left(\left(\frac{a \cdot c}{{b}^{3}} \cdot c + \frac{1}{b} \cdot c\right)\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto \mathsf{neg}\left(c \cdot \left(\frac{a \cdot c}{{b}^{3}} + \frac{1}{b}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{neg}\left(c \cdot \left(\frac{1}{b} + \frac{a \cdot c}{{b}^{3}}\right)\right) \]

   11. distribute-rgt-neg-in N/A

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

   12. +-commutative N/A

       \[\leadsto c \cdot \left(\mathsf{neg}\left(\left(\frac{a \cdot c}{{b}^{3}} + \frac{1}{b}\right)\right)\right) \]

   13. distribute-neg-out N/A

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

   14. mul-1-neg N/A

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

   15. sub-neg N/A

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

10. Simplified 93.4%

    \[\leadsto \color{blue}{c \cdot \frac{-1 - a \cdot \frac{c}{b \cdot b}}{b}} \]
11. Add Preprocessing

Alternative 9: 90.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{c}{b} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 - (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 = 0.0d0 - (c / b)
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 - Float64(c / b))
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

   4. /-lowering-/.f64 88.8%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

5. Simplified 88.8%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

   3. /-lowering-/.f64 88.8%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

7. Applied egg-rr 88.8%

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

8. Final simplification 88.8%

   \[\leadsto 0 - \frac{c}{b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))