Quadratic roots, wide range

Percentage Accurate: 18.0% → 97.5%

Time: 15.0s

Alternatives: 8

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: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{\frac{a}{\frac{a}{\frac{-2 \cdot \left(c + \frac{a \cdot c}{\frac{b}{\frac{c}{b}}}\right)}{b} + \left(a \cdot a\right) \cdot \left(\frac{\left(c \cdot \left(\left(a \cdot c\right) \cdot -0.5\right)\right) \cdot \left(c \cdot \left(c \cdot 20\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot t\_0\right)\right)} + \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-4}{b \cdot b}}{t\_0}\right)}}}{2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (/
    (/
     a
     (/
      a
      (+
       (/ (* -2.0 (+ c (/ (* a c) (/ b (/ c b))))) b)
       (*
        (* a a)
        (+
         (/
          (* (* c (* (* a c) -0.5)) (* c (* c 20.0)))
          (* (* b b) (* b (* b t_0))))
         (/ (* (* c (* c c)) (/ -4.0 (* b b))) t_0))))))
    2.0)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (a / (a / (((-2.0 * (c + ((a * c) / (b / (c / b))))) / b) + ((a * a) * ((((c * ((a * c) * -0.5)) * (c * (c * 20.0))) / ((b * b) * (b * (b * t_0)))) + (((c * (c * c)) * (-4.0 / (b * b))) / t_0)))))) / 2.0;
}

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 / ((((-2.0d0) * (c + ((a * c) / (b / (c / b))))) / b) + ((a * a) * ((((c * ((a * c) * (-0.5d0))) * (c * (c * 20.0d0))) / ((b * b) * (b * (b * t_0)))) + (((c * (c * c)) * ((-4.0d0) / (b * b))) / t_0)))))) / 2.0d0
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (a / (a / (((-2.0 * (c + ((a * c) / (b / (c / b))))) / b) + ((a * a) * ((((c * ((a * c) * -0.5)) * (c * (c * 20.0))) / ((b * b) * (b * (b * t_0)))) + (((c * (c * c)) * (-4.0 / (b * b))) / t_0)))))) / 2.0;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return (a / (a / (((-2.0 * (c + ((a * c) / (b / (c / b))))) / b) + ((a * a) * ((((c * ((a * c) * -0.5)) * (c * (c * 20.0))) / ((b * b) * (b * (b * t_0)))) + (((c * (c * c)) * (-4.0 / (b * b))) / t_0)))))) / 2.0

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(a / Float64(a / Float64(Float64(Float64(-2.0 * Float64(c + Float64(Float64(a * c) / Float64(b / Float64(c / b))))) / b) + Float64(Float64(a * a) * Float64(Float64(Float64(Float64(c * Float64(Float64(a * c) * -0.5)) * Float64(c * Float64(c * 20.0))) / Float64(Float64(b * b) * Float64(b * Float64(b * t_0)))) + Float64(Float64(Float64(c * Float64(c * c)) * Float64(-4.0 / Float64(b * b))) / t_0)))))) / 2.0)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = (a / (a / (((-2.0 * (c + ((a * c) / (b / (c / b))))) / b) + ((a * a) * ((((c * ((a * c) * -0.5)) * (c * (c * 20.0))) / ((b * b) * (b * (b * t_0)))) + (((c * (c * c)) * (-4.0 / (b * b))) / t_0)))))) / 2.0;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(a / N[(a / N[(N[(N[(-2.0 * N[(c + N[(N[(a * c), $MachinePrecision] / N[(b / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(N[(N[(c * N[(N[(a * c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(c * N[(c * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{2} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)\right)}, \mathsf{*.f64}\left(2, a\right)\right) \]

5. Simplified 97.5%

   \[\leadsto \frac{\color{blue}{a \cdot \left(\frac{c \cdot -2}{b} + a \cdot \left(\frac{-2 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\frac{-4 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}} + \frac{\left(-0.5 \cdot a\right) \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}}{b}\right)\right)\right)}}{2 \cdot a} \]

7. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{\frac{\frac{a \cdot \left(-2 \cdot \left(\frac{1}{b} \cdot \left(c + \frac{a \cdot c}{\frac{b}{\frac{c}{b}}}\right)\right) + \left(\frac{-4 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot -0.5\right) \cdot \frac{\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot \left(a \cdot a\right)\right)}{a}}{2}} \]

9. Applied egg-rr 97.5%

   \[\leadsto \frac{\color{blue}{\frac{\frac{-2 \cdot \left(c + \frac{c \cdot a}{\frac{b}{\frac{c}{b}}}\right)}{b} + \left(\frac{\frac{-4 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)} + \frac{\left(a \cdot -0.5\right) \cdot \left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right) \cdot \left(a \cdot a\right)}{a} \cdot a}}{2} \]

11. Applied egg-rr 97.5%

    \[\leadsto \frac{\color{blue}{\frac{a}{\frac{a}{\frac{-2 \cdot \left(c + \frac{c \cdot a}{\frac{b}{\frac{c}{b}}}\right)}{b} + \left(a \cdot a\right) \cdot \left(\frac{\left(\left(-0.5 \cdot \left(c \cdot a\right)\right) \cdot c\right) \cdot \left(c \cdot \left(c \cdot 20\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} + \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-4}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}}}}{2} \]

12. Final simplification 97.5%

    \[\leadsto \frac{\frac{a}{\frac{a}{\frac{-2 \cdot \left(c + \frac{a \cdot c}{\frac{b}{\frac{c}{b}}}\right)}{b} + \left(a \cdot a\right) \cdot \left(\frac{\left(c \cdot \left(\left(a \cdot c\right) \cdot -0.5\right)\right) \cdot \left(c \cdot \left(c \cdot 20\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} + \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-4}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}}}{2} \]
13. Add Preprocessing

Alternative 2: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{a \cdot \frac{\frac{-2 \cdot \left(c + \frac{a \cdot c}{\frac{b}{\frac{c}{b}}}\right)}{b} + \left(a \cdot a\right) \cdot \left(\frac{\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -4}{b \cdot b}}{t\_0} + \frac{\left(a \cdot -0.5\right) \cdot \left(c \cdot \left(c \cdot \left(20 \cdot \left(c \cdot c\right)\right)\right)\right)}{b \cdot \left(t\_0 \cdot t\_0\right)}\right)}{a}}{2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (/
    (*
     a
     (/
      (+
       (/ (* -2.0 (+ c (/ (* a c) (/ b (/ c b))))) b)
       (*
        (* a a)
        (+
         (/ (/ (* (* c (* c c)) -4.0) (* b b)) t_0)
         (/ (* (* a -0.5) (* c (* c (* 20.0 (* c c))))) (* b (* t_0 t_0))))))
      a))
    2.0)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (a * ((((-2.0 * (c + ((a * c) / (b / (c / b))))) / b) + ((a * a) * (((((c * (c * c)) * -4.0) / (b * b)) / t_0) + (((a * -0.5) * (c * (c * (20.0 * (c * c))))) / (b * (t_0 * t_0)))))) / a)) / 2.0;
}

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 * (((((-2.0d0) * (c + ((a * c) / (b / (c / b))))) / b) + ((a * a) * (((((c * (c * c)) * (-4.0d0)) / (b * b)) / t_0) + (((a * (-0.5d0)) * (c * (c * (20.0d0 * (c * c))))) / (b * (t_0 * t_0)))))) / a)) / 2.0d0
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (a * ((((-2.0 * (c + ((a * c) / (b / (c / b))))) / b) + ((a * a) * (((((c * (c * c)) * -4.0) / (b * b)) / t_0) + (((a * -0.5) * (c * (c * (20.0 * (c * c))))) / (b * (t_0 * t_0)))))) / a)) / 2.0;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return (a * ((((-2.0 * (c + ((a * c) / (b / (c / b))))) / b) + ((a * a) * (((((c * (c * c)) * -4.0) / (b * b)) / t_0) + (((a * -0.5) * (c * (c * (20.0 * (c * c))))) / (b * (t_0 * t_0)))))) / a)) / 2.0

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(a * Float64(Float64(Float64(Float64(-2.0 * Float64(c + Float64(Float64(a * c) / Float64(b / Float64(c / b))))) / b) + Float64(Float64(a * a) * Float64(Float64(Float64(Float64(Float64(c * Float64(c * c)) * -4.0) / Float64(b * b)) / t_0) + Float64(Float64(Float64(a * -0.5) * Float64(c * Float64(c * Float64(20.0 * Float64(c * c))))) / Float64(b * Float64(t_0 * t_0)))))) / a)) / 2.0)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = (a * ((((-2.0 * (c + ((a * c) / (b / (c / b))))) / b) + ((a * a) * (((((c * (c * c)) * -4.0) / (b * b)) / t_0) + (((a * -0.5) * (c * (c * (20.0 * (c * c))))) / (b * (t_0 * t_0)))))) / a)) / 2.0;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(N[(N[(N[(-2.0 * N[(c + N[(N[(a * c), $MachinePrecision] / N[(b / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[(a * -0.5), $MachinePrecision] * N[(c * N[(c * N[(20.0 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{2} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)\right)}, \mathsf{*.f64}\left(2, a\right)\right) \]

5. Simplified 97.5%

   \[\leadsto \frac{\color{blue}{a \cdot \left(\frac{c \cdot -2}{b} + a \cdot \left(\frac{-2 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\frac{-4 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}} + \frac{\left(-0.5 \cdot a\right) \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}}{b}\right)\right)\right)}}{2 \cdot a} \]

7. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{\frac{\frac{a \cdot \left(-2 \cdot \left(\frac{1}{b} \cdot \left(c + \frac{a \cdot c}{\frac{b}{\frac{c}{b}}}\right)\right) + \left(\frac{-4 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot -0.5\right) \cdot \frac{\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot \left(a \cdot a\right)\right)}{a}}{2}} \]

9. Applied egg-rr 97.5%

   \[\leadsto \frac{\color{blue}{\frac{\frac{-2 \cdot \left(c + \frac{c \cdot a}{\frac{b}{\frac{c}{b}}}\right)}{b} + \left(\frac{\frac{-4 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)} + \frac{\left(a \cdot -0.5\right) \cdot \left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right) \cdot \left(a \cdot a\right)}{a} \cdot a}}{2} \]

10. Final simplification 97.5%

    \[\leadsto \frac{a \cdot \frac{\frac{-2 \cdot \left(c + \frac{a \cdot c}{\frac{b}{\frac{c}{b}}}\right)}{b} + \left(a \cdot a\right) \cdot \left(\frac{\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -4}{b \cdot b}}{b \cdot \left(b \cdot b\right)} + \frac{\left(a \cdot -0.5\right) \cdot \left(c \cdot \left(c \cdot \left(20 \cdot \left(c \cdot c\right)\right)\right)\right)}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right)}{a}}{2} \]
11. Add Preprocessing

Alternative 3: 96.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(0 - 2\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} - \frac{c \cdot \left(a \cdot c\right)}{b \cdot b}\right) - c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (-
    (/ (* (* a a) (* (* c (* c c)) (- 0.0 2.0))) (* (* b b) (* b b)))
    (/ (* c (* a c)) (* b b)))
   c)
  b))

C

double code(double a, double b, double c) {
	return (((((a * a) * ((c * (c * c)) * (0.0 - 2.0))) / ((b * b) * (b * b))) - ((c * (a * c)) / (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 * a) * ((c * (c * c)) * (0.0d0 - 2.0d0))) / ((b * b) * (b * b))) - ((c * (a * c)) / (b * b))) - c) / b
end function

Java

public static double code(double a, double b, double c) {
	return (((((a * a) * ((c * (c * c)) * (0.0 - 2.0))) / ((b * b) * (b * b))) - ((c * (a * c)) / (b * b))) - c) / b;
}

Python

def code(a, b, c):
	return (((((a * a) * ((c * (c * c)) * (0.0 - 2.0))) / ((b * b) * (b * b))) - ((c * (a * c)) / (b * b))) - c) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64(a * a) * Float64(Float64(c * Float64(c * c)) * Float64(0.0 - 2.0))) / Float64(Float64(b * b) * Float64(b * b))) - Float64(Float64(c * Float64(a * c)) / Float64(b * b))) - c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((((a * a) * ((c * (c * c)) * (0.0 - 2.0))) / ((b * b) * (b * b))) - ((c * (a * c)) / (b * b))) - c) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(0.0 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{2} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)\right)}, \mathsf{*.f64}\left(2, a\right)\right) \]

5. Simplified 97.5%

   \[\leadsto \frac{\color{blue}{a \cdot \left(\frac{c \cdot -2}{b} + a \cdot \left(\frac{-2 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\frac{-4 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}} + \frac{\left(-0.5 \cdot a\right) \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}}{b}\right)\right)\right)}}{2 \cdot a} \]

7. Applied egg-rr 97.5%

   \[\leadsto \frac{a \cdot \color{blue}{\left(\frac{c \cdot -2}{b} + a \cdot \left(\frac{c \cdot \left(c \cdot -2\right)}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\frac{-4 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot -0.5\right) \cdot \frac{\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right)\right)\right)}}{2 \cdot a} \]

8. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

10. Simplified 97.1%

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

11. Final simplification 97.1%

    \[\leadsto \frac{\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(0 - 2\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} - \frac{c \cdot \left(a \cdot c\right)}{b \cdot b}\right) - c}{b} \]
12. Add Preprocessing

Alternative 4: 96.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(-2.0 * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $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 16.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 97.1%

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

6. Final simplification 97.1%

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

Alternative 5: 96.5% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in c around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(c, \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \mathsf{*.f64}\left(c, \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) + \frac{\mathsf{neg}\left(1\right)}{\color{blue}{b}}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(c, \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right)\right), \color{blue}{\left(\frac{-1}{b}\right)}\right)\right) \]

5. Simplified 96.8%

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

6. Taylor expanded in b around inf

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

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

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

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

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-*r/ N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

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

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

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

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

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

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({a}^{2}\right)\right), b\right), b\right)\right), a\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(-1, b\right)\right)\right) \]

   15. unpow2 N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(a, a\right)\right), b\right), b\right)\right), a\right), \left({b}^{3}\right)\right)\right), \mathsf{/.f64}\left(-1, b\right)\right)\right) \]

   17. cube-mult N/A

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

8. Simplified 96.8%

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

Alternative 6: 95.2% 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 16.5%

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

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{2} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)\right)}, \mathsf{*.f64}\left(2, a\right)\right) \]

5. Simplified 97.5%

   \[\leadsto \frac{\color{blue}{a \cdot \left(\frac{c \cdot -2}{b} + a \cdot \left(\frac{-2 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\frac{-4 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}} + \frac{\left(-0.5 \cdot a\right) \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}}{b}\right)\right)\right)}}{2 \cdot a} \]

6. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. associate-*r/ N/A

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

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. div-sub N/A

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

   8. unsub-neg N/A

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

   9. mul-1-neg N/A

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

   10. distribute-lft-out N/A

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

   11. associate-*r/ N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. /-lowering-/.f64 N/A

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

8. Simplified 95.6%

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

9. Final simplification 95.6%

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

Alternative 7: 95.1% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / ((a / b) - (b / c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in a around 0

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

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

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

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

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

   9. unpow3 N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

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

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

5. Simplified 95.5%

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

6. Taylor expanded in a around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. +-commutative N/A

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

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

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

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

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

   10. unpow2 N/A

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

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

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

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

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

   15. unpow2 N/A

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

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

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

   17. /-lowering-/.f64 N/A

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

   18. *-commutative N/A

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

   19. *-lowering-*.f64 94.9%

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

8. Simplified 94.9%

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

   1. associate-/r* N/A

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

   2. clear-num N/A

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

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

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

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

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

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

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

10. Applied egg-rr 94.8%

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

11. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. unsub-neg N/A

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

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

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

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

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

    6. /-lowering-/.f64 95.4%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{/.f64}\left(b, \color{blue}{c}\right)\right)\right) \]

13. Simplified 95.4%

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

Alternative 8: 90.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 91.2%

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

5. Simplified 91.2%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
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 91.2%

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

7. Applied egg-rr 91.2%

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

8. Final simplification 91.2%

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

Reproduce

herbie shell --seed 2024191 
(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)))