Quadratic roots, medium range

Percentage Accurate: 32.2% → 95.5%

Time: 15.3s

Alternatives: 9

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 32.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\\ t_2 := b \cdot \left(b \cdot \left(b \cdot \left(a \cdot t\_0\right)\right)\right)\\ t_3 := \frac{t\_1}{t\_2}\\ \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \frac{\mathsf{fma}\left(0.0625, t\_3 \cdot t\_3, \mathsf{fma}\left(-c, c, \frac{\left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot c\right)}{b \cdot t\_0}\right)\right)}{\frac{t\_1 \cdot -0.25}{t\_2} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, -c\right)}\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (* a (* (* a (* a a)) (* (* c c) (* (* c c) 20.0)))))
        (t_2 (* b (* b (* b (* a t_0)))))
        (t_3 (/ t_1 t_2)))
   (/
    (fma
     (/ (* a (* c (* a (* c c)))) (* (* b b) (* b b)))
     -2.0
     (/
      (fma
       0.0625
       (* t_3 t_3)
       (fma (- c) c (/ (* (* a (* c (* c c))) (* a c)) (* b t_0))))
      (- (/ (* t_1 -0.25) t_2) (fma a (/ (* c c) (* b b)) (- c)))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = a * ((a * (a * a)) * ((c * c) * ((c * c) * 20.0)));
	double t_2 = b * (b * (b * (a * t_0)));
	double t_3 = t_1 / t_2;
	return fma(((a * (c * (a * (c * c)))) / ((b * b) * (b * b))), -2.0, (fma(0.0625, (t_3 * t_3), fma(-c, c, (((a * (c * (c * c))) * (a * c)) / (b * t_0)))) / (((t_1 * -0.25) / t_2) - fma(a, ((c * c) / (b * b)), -c)))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(a * Float64(Float64(a * Float64(a * a)) * Float64(Float64(c * c) * Float64(Float64(c * c) * 20.0))))
	t_2 = Float64(b * Float64(b * Float64(b * Float64(a * t_0))))
	t_3 = Float64(t_1 / t_2)
	return Float64(fma(Float64(Float64(a * Float64(c * Float64(a * Float64(c * c)))) / Float64(Float64(b * b) * Float64(b * b))), -2.0, Float64(fma(0.0625, Float64(t_3 * t_3), fma(Float64(-c), c, Float64(Float64(Float64(a * Float64(c * Float64(c * c))) * Float64(a * c)) / Float64(b * t_0)))) / Float64(Float64(Float64(t_1 * -0.25) / t_2) - fma(a, Float64(Float64(c * c) / Float64(b * b)), Float64(-c))))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * N[(b * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, N[(N[(N[(N[(a * N[(c * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(0.0625 * N[(t$95$3 * t$95$3), $MachinePrecision] + N[((-c) * c + N[(N[(N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$1 * -0.25), $MachinePrecision] / t$95$2), $MachinePrecision] - N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \mathsf{fma}\left(-0.25, \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 20\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}, -\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)\right)\right)}{b} \]

9. Applied rewrites 95.7%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \frac{\mathsf{fma}\left(0.0625, \frac{a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)} \cdot \frac{a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \mathsf{fma}\left(c, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, -c \cdot c\right)\right)}{\frac{-0.25 \cdot \left(a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, -c\right)}\right)}{b} \]
10. Step-by-step derivation

11. Applied rewrites 95.8%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \frac{\mathsf{fma}\left(0.0625, \frac{a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)} \cdot \frac{a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \mathsf{fma}\left(-c, c, \frac{\left(a \cdot c\right) \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{\frac{-0.25 \cdot \left(a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, -c\right)}\right)}{b} \]

12. Final simplification 95.8%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \frac{\mathsf{fma}\left(0.0625, \frac{a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)} \cdot \frac{a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \mathsf{fma}\left(-c, c, \frac{\left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot c\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{\frac{\left(a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\right) \cdot -0.25}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, -c\right)}\right)}{b} \]
13. Add Preprocessing

Alternative 2: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot t\_0\\ t_2 := a \cdot \left(a \cdot a\right)\\ t_3 := \left(b \cdot b\right) \cdot \left(a \cdot t\_1\right)\\ t_4 := a \cdot \left(c \cdot c\right)\\ t_5 := \left(c \cdot c\right) \cdot 20\\ t_6 := t\_2 \cdot \left(c \cdot \left(c \cdot t\_5\right)\right)\\ \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot t\_4\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \frac{\mathsf{fma}\left(0.0625, \frac{t\_6 \cdot \left(a \cdot \left(a \cdot t\_6\right)\right)}{t\_3 \cdot t\_3}, \mathsf{fma}\left(c, \frac{c \cdot \left(a \cdot t\_4\right)}{t\_1}, c \cdot \left(-c\right)\right)\right)}{\frac{\left(a \cdot \left(t\_2 \cdot \left(\left(c \cdot c\right) \cdot t\_5\right)\right)\right) \cdot -0.25}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot t\_0\right)\right)\right)} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, -c\right)}\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (* b t_0))
        (t_2 (* a (* a a)))
        (t_3 (* (* b b) (* a t_1)))
        (t_4 (* a (* c c)))
        (t_5 (* (* c c) 20.0))
        (t_6 (* t_2 (* c (* c t_5)))))
   (/
    (fma
     (/ (* a (* c t_4)) (* (* b b) (* b b)))
     -2.0
     (/
      (fma
       0.0625
       (/ (* t_6 (* a (* a t_6))) (* t_3 t_3))
       (fma c (/ (* c (* a t_4)) t_1) (* c (- c))))
      (-
       (/ (* (* a (* t_2 (* (* c c) t_5))) -0.25) (* b (* b (* b (* a t_0)))))
       (fma a (/ (* c c) (* b b)) (- c)))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * t_0;
	double t_2 = a * (a * a);
	double t_3 = (b * b) * (a * t_1);
	double t_4 = a * (c * c);
	double t_5 = (c * c) * 20.0;
	double t_6 = t_2 * (c * (c * t_5));
	return fma(((a * (c * t_4)) / ((b * b) * (b * b))), -2.0, (fma(0.0625, ((t_6 * (a * (a * t_6))) / (t_3 * t_3)), fma(c, ((c * (a * t_4)) / t_1), (c * -c))) / ((((a * (t_2 * ((c * c) * t_5))) * -0.25) / (b * (b * (b * (a * t_0))))) - fma(a, ((c * c) / (b * b)), -c)))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(b * t_0)
	t_2 = Float64(a * Float64(a * a))
	t_3 = Float64(Float64(b * b) * Float64(a * t_1))
	t_4 = Float64(a * Float64(c * c))
	t_5 = Float64(Float64(c * c) * 20.0)
	t_6 = Float64(t_2 * Float64(c * Float64(c * t_5)))
	return Float64(fma(Float64(Float64(a * Float64(c * t_4)) / Float64(Float64(b * b) * Float64(b * b))), -2.0, Float64(fma(0.0625, Float64(Float64(t_6 * Float64(a * Float64(a * t_6))) / Float64(t_3 * t_3)), fma(c, Float64(Float64(c * Float64(a * t_4)) / t_1), Float64(c * Float64(-c)))) / Float64(Float64(Float64(Float64(a * Float64(t_2 * Float64(Float64(c * c) * t_5))) * -0.25) / Float64(b * Float64(b * Float64(b * Float64(a * t_0))))) - fma(a, Float64(Float64(c * c) / Float64(b * b)), Float64(-c))))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b * b), $MachinePrecision] * N[(a * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(c * c), $MachinePrecision] * 20.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 * N[(c * N[(c * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(a * N[(c * t$95$4), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(0.0625 * N[(N[(t$95$6 * N[(a * N[(a * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(c * N[(a * t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(c * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(a * N[(t$95$2 * N[(N[(c * c), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / N[(b * N[(b * N[(b * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \mathsf{fma}\left(-0.25, \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 20\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}, -\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)\right)\right)}{b} \]

9. Applied rewrites 95.7%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \frac{\mathsf{fma}\left(0.0625, \frac{a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)} \cdot \frac{a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \mathsf{fma}\left(c, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, -c \cdot c\right)\right)}{\frac{-0.25 \cdot \left(a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, -c\right)}\right)}{b} \]
10. Step-by-step derivation

11. Applied rewrites 95.7%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \frac{\mathsf{fma}\left(0.0625, \frac{\left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\right) \cdot \left(a \cdot \left(a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\right)\right)\right)}{\left(\left(b \cdot b\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \mathsf{fma}\left(c, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, -c \cdot c\right)\right)}{\frac{-0.25 \cdot \left(a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, -c\right)}\right)}{b} \]

12. Final simplification 95.7%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \frac{\mathsf{fma}\left(0.0625, \frac{\left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\right) \cdot \left(a \cdot \left(a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\right)\right)\right)}{\left(\left(b \cdot b\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, \mathsf{fma}\left(c, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c \cdot \left(-c\right)\right)\right)}{\frac{\left(a \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)\right) \cdot -0.25}{b \cdot \left(b \cdot \left(b \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, -c\right)}\right)}{b} \]
13. Add Preprocessing

Alternative 3: 95.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \mathsf{fma}\left(-0.25, \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 20\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}, -\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)\right)\right)}{b} \]

8. Final simplification 95.5%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \mathsf{fma}\left(-0.25, \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(20 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, -\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)\right)\right)}{b} \]
9. Add Preprocessing

Alternative 4: 95.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

   \[\leadsto \frac{\mathsf{fma}\left(-0.25, \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 20\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]

8. Final simplification 95.5%

   \[\leadsto \frac{\mathsf{fma}\left(-0.25, \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(20 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \left(c \cdot -2\right) \cdot \frac{c \cdot \left(a \cdot \left(a \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]
9. Add Preprocessing

Alternative 5: 93.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. lower-/.f64 N/A

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

5. Applied rewrites 93.5%

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

7. Applied rewrites 93.5%

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

8. Final simplification 93.5%

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

Alternative 6: 93.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. lower-/.f64 N/A

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

5. Applied rewrites 93.5%

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

Alternative 7: 90.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 89.5

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

5. Applied rewrites 89.5%

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

6. Final simplification 89.5%

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

Alternative 8: 80.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 77.9

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

5. Applied rewrites 77.9%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Alternative 9: 3.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0;
}

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
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 36.8%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 36.3%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. metadata-eval N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 37.4

       \[\leadsto \mathsf{fma}\left(b, -\color{blue}{\frac{0.5}{a}}, \frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}{a \cdot 2}\right) \]

6. Applied rewrites 37.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b, -\frac{0.5}{a}, \frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{a \cdot 2}\right)} \]

7. Taylor expanded in a around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{b \cdot \color{blue}{0}}{a} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{b}{a} \cdot 0} \]

   4. mul0-rgt 3.2

      \[\leadsto \color{blue}{0} \]

9. Applied rewrites 3.2%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))