Cubic critical, medium range

Percentage Accurate: 32.2% → 95.3%

Time: 15.0s

Alternatives: 9

Speedup: 2.9×

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

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Initial Program: 32.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Alternative 1: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, \frac{-0.16666666666666666 \cdot \left(\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{t\_0 \cdot \left(a \cdot t\_0\right)}\right) + \left(a \cdot a\right) \cdot \frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{b \cdot t\_0}\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (/
    (fma
     c
     -0.5
     (+
      (fma
       a
       (/ (* (* c c) -0.375) (* b b))
       (/
        (*
         -0.16666666666666666
         (* (* a (* a (* a a))) (* (* c c) (* (* c c) 6.328125))))
        (* t_0 (* a t_0))))
      (* (* a a) (/ (* c (* (* c c) -0.5625)) (* b t_0)))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma(c, -0.5, (fma(a, (((c * c) * -0.375) / (b * b)), ((-0.16666666666666666 * ((a * (a * (a * a))) * ((c * c) * ((c * c) * 6.328125)))) / (t_0 * (a * t_0)))) + ((a * a) * ((c * ((c * c) * -0.5625)) / (b * t_0))))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(fma(c, -0.5, Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(Float64(-0.16666666666666666 * Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(Float64(c * c) * Float64(Float64(c * c) * 6.328125)))) / Float64(t_0 * Float64(a * t_0)))) + Float64(Float64(a * a) * Float64(Float64(c * Float64(Float64(c * c) * -0.5625)) / Float64(b * t_0))))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * -0.5 + N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]

7. Applied rewrites 95.9%

   \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, \frac{-0.16666666666666666 \cdot \left(\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}\right) + \left(a \cdot a\right) \cdot \frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)}{b} \]

8. Final simplification 95.9%

   \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, \frac{-0.16666666666666666 \cdot \left(\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right) + \left(a \cdot a\right) \cdot \frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)}{b} \]
9. Add Preprocessing

Alternative 2: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot c\right)\\ t_1 := \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot \left(6.328125 \cdot \left(c \cdot t\_0\right)\right)\right)}{\left(b \cdot b\right) \cdot t\_1}, -0.16666666666666666, a \cdot \mathsf{fma}\left(-0.5625 \cdot t\_0, \frac{a}{t\_1}, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* c c))) (t_1 (* (* b b) (* b b))))
   (/
    (fma
     c
     -0.5
     (fma
      (/ (* (* a a) (* a (* 6.328125 (* c t_0)))) (* (* b b) t_1))
      -0.16666666666666666
      (* a (fma (* -0.5625 t_0) (/ a t_1) (/ (* (* c c) -0.375) (* b b))))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = (b * b) * (b * b);
	return fma(c, -0.5, fma((((a * a) * (a * (6.328125 * (c * t_0)))) / ((b * b) * t_1)), -0.16666666666666666, (a * fma((-0.5625 * t_0), (a / t_1), (((c * c) * -0.375) / (b * b)))))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(c * c))
	t_1 = Float64(Float64(b * b) * Float64(b * b))
	return Float64(fma(c, -0.5, fma(Float64(Float64(Float64(a * a) * Float64(a * Float64(6.328125 * Float64(c * t_0)))) / Float64(Float64(b * b) * t_1)), -0.16666666666666666, Float64(a * fma(Float64(-0.5625 * t_0), Float64(a / t_1), Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)))))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * -0.5 + N[(N[(N[(N[(a * a), $MachinePrecision] * N[(a * N[(6.328125 * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(a * N[(N[(-0.5625 * t$95$0), $MachinePrecision] * N[(a / t$95$1), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]

7. Applied rewrites 95.9%

   \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, \frac{-0.16666666666666666 \cdot \left(\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}\right) + \left(a \cdot a\right) \cdot \frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)}{b} \]

9. Applied rewrites 95.9%

   \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, -0.16666666666666666, \mathsf{fma}\left(a, \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot b}, \frac{a \cdot \left(a \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)\right)}{b} \]

11. Applied rewrites 95.9%

    \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 6.328125\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}, -0.16666666666666666, a \cdot \mathsf{fma}\left(-0.5625 \cdot \left(c \cdot \left(c \cdot c\right)\right), \frac{a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)\right)}{b} \]

12. Final simplification 95.9%

    \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot \left(6.328125 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}, -0.16666666666666666, a \cdot \mathsf{fma}\left(-0.5625 \cdot \left(c \cdot \left(c \cdot c\right)\right), \frac{a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \]
13. Add Preprocessing

Alternative 3: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot c\right)\\ t_1 := \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.5625 \cdot t\_0, \frac{a}{t\_1}, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot \left(6.328125 \cdot \left(c \cdot t\_0\right)\right)\right)}{\left(b \cdot b\right) \cdot t\_1}, -0.16666666666666666, c \cdot -0.5\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* c c))) (t_1 (* (* b b) (* b b))))
   (/
    (fma
     a
     (fma (* -0.5625 t_0) (/ a t_1) (/ (* (* c c) -0.375) (* b b)))
     (fma
      (/ (* (* a a) (* a (* 6.328125 (* c t_0)))) (* (* b b) t_1))
      -0.16666666666666666
      (* c -0.5)))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = (b * b) * (b * b);
	return fma(a, fma((-0.5625 * t_0), (a / t_1), (((c * c) * -0.375) / (b * b))), fma((((a * a) * (a * (6.328125 * (c * t_0)))) / ((b * b) * t_1)), -0.16666666666666666, (c * -0.5))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(c * c))
	t_1 = Float64(Float64(b * b) * Float64(b * b))
	return Float64(fma(a, fma(Float64(-0.5625 * t_0), Float64(a / t_1), Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b))), fma(Float64(Float64(Float64(a * a) * Float64(a * Float64(6.328125 * Float64(c * t_0)))) / Float64(Float64(b * b) * t_1)), -0.16666666666666666, Float64(c * -0.5))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(N[(-0.5625 * t$95$0), $MachinePrecision] * N[(a / t$95$1), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * N[(a * N[(6.328125 * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(c * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]

7. Applied rewrites 95.9%

   \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, \frac{-0.16666666666666666 \cdot \left(\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}\right) + \left(a \cdot a\right) \cdot \frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)}{b} \]

9. Applied rewrites 95.9%

   \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, -0.16666666666666666, \mathsf{fma}\left(a, \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot b}, \frac{a \cdot \left(a \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)\right)}{b} \]

11. Applied rewrites 95.9%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.5625 \cdot \left(c \cdot \left(c \cdot c\right)\right), \frac{a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}\right), \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 6.328125\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}, -0.16666666666666666, c \cdot -0.5\right)\right)}{b}} \]

12. Final simplification 95.9%

    \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.5625 \cdot \left(c \cdot \left(c \cdot c\right)\right), \frac{a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot \left(6.328125 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}, -0.16666666666666666, c \cdot -0.5\right)\right)}{b} \]
13. Add Preprocessing

Alternative 4: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c, -0.5, \frac{\mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}, -0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   c
   -0.5
   (/
    (fma
     -0.5625
     (/ (* (* a a) (* c (* c c))) (* b b))
     (* -0.375 (* a (* c c))))
    (* b b)))
  b))

C

double code(double a, double b, double c) {
	return fma(c, -0.5, (fma(-0.5625, (((a * a) * (c * (c * c))) / (b * b)), (-0.375 * (a * (c * c)))) / (b * b))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(c, -0.5, Float64(fma(-0.5625, Float64(Float64(Float64(a * a) * Float64(c * Float64(c * c))) / Float64(b * b)), Float64(-0.375 * Float64(a * Float64(c * c)))) / Float64(b * b))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * -0.5 + N[(N[(-0.5625 * N[(N[(N[(a * a), $MachinePrecision] * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]

7. Applied rewrites 95.9%

   \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, \frac{-0.16666666666666666 \cdot \left(\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}\right) + \left(a \cdot a\right) \cdot \frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)}{b} \]

8. Taylor expanded in b around inf

   \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right)}{b} \]
9. Step-by-step derivation

10. Applied rewrites 94.2%

    \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \frac{\mathsf{fma}\left(-0.5625, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{b \cdot b}, -0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b} \]

11. Final simplification 94.2%

    \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \frac{\mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}, -0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b} \]
12. Add Preprocessing

Alternative 5: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-0.5625, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (fma
   c
   (/ (fma -0.5625 (/ (* c (* a a)) (* b b)) (* a -0.375)) (* b (* b b)))
   (/ -0.5 b))))

C

double code(double a, double b, double c) {
	return c * fma(c, (fma(-0.5625, ((c * (a * a)) / (b * b)), (a * -0.375)) / (b * (b * b))), (-0.5 / b));
}

Julia

function code(a, b, c)
	return Float64(c * fma(c, Float64(fma(-0.5625, Float64(Float64(c * Float64(a * a)) / Float64(b * b)), Float64(a * -0.375)) / Float64(b * Float64(b * b))), Float64(-0.5 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(c * N[(N[(-0.5625 * N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

   2. sub-neg N/A

      \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

   3. associate-*r/ N/A

      \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right)}{{b}^{5}}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto c \cdot \left(c \cdot \left(\frac{\color{blue}{\left(\frac{-9}{16} \cdot {a}^{2}\right) \cdot c}}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{\frac{-9}{16} \cdot {a}^{2}}{{b}^{5}} \cdot c} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right)} \cdot c + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, \frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

5. Applied rewrites 93.9%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(a \cdot a, \frac{c}{{b}^{5}} \cdot -0.5625, \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{\color{blue}{{b}^{3}}}, \frac{\frac{-1}{2}}{b}\right) \]
7. Step-by-step derivation

8. Applied rewrites 93.9%

   \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-0.5625, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -0.375 \cdot a\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-0.5}{b}\right) \]

9. Final simplification 93.9%

   \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-0.5625, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right) \]
10. Add Preprocessing

Alternative 6: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b))

C

double code(double a, double b, double c) {
	return fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

5. Applied rewrites 90.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
6. Add Preprocessing

Alternative 7: 90.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot -0.375, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (fma (/ a (* b b)) (* c -0.375) -0.5)) b))

C

double code(double a, double b, double c) {
	return (c * fma((a / (b * b)), (c * -0.375), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(Float64(a / Float64(b * b)), Float64(c * -0.375), -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * -0.375), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

8. Applied rewrites 90.5%

   \[\leadsto \color{blue}{\frac{c \cdot \mathsf{fma}\left(\frac{a}{b \cdot b}, -0.375 \cdot c, -0.5\right)}{b}} \]

9. Final simplification 90.5%

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot -0.375, -0.5\right)}{b} \]
10. Add Preprocessing

Alternative 8: 90.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{\mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot -0.375, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ (fma (/ a (* b b)) (* c -0.375) -0.5) b)))

C

double code(double a, double b, double c) {
	return c * (fma((a / (b * b)), (c * -0.375), -0.5) / b);
}

Julia

function code(a, b, c)
	return Float64(c * Float64(fma(Float64(a / Float64(b * b)), Float64(c * -0.375), -0.5) / b))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * -0.375), $MachinePrecision] + -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]

6. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

   2. associate-*r/ N/A

      \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} - \frac{1}{2} \cdot \frac{1}{b}\right) \]

   3. unpow3 N/A

      \[\leadsto c \cdot \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{\color{blue}{\left(b \cdot b\right) \cdot b}} - \frac{1}{2} \cdot \frac{1}{b}\right) \]

   4. unpow2 N/A

      \[\leadsto c \cdot \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{\color{blue}{{b}^{2}} \cdot b} - \frac{1}{2} \cdot \frac{1}{b}\right) \]

   5. associate-/r* N/A

      \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}}}{b}} - \frac{1}{2} \cdot \frac{1}{b}\right) \]

   6. associate-*r/ N/A

      \[\leadsto c \cdot \left(\frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}}}}{b} - \frac{1}{2} \cdot \frac{1}{b}\right) \]

   7. associate-*r/ N/A

      \[\leadsto c \cdot \left(\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}}}{b} - \color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}\right) \]

   8. metadata-eval N/A

      \[\leadsto c \cdot \left(\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}}}{b} - \frac{\color{blue}{\frac{1}{2}}}{b}\right) \]

   9. div-sub N/A

      \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b}} \]

8. Applied rewrites 90.3%

   \[\leadsto \color{blue}{c \cdot \frac{\mathsf{fma}\left(\frac{a}{b \cdot b}, -0.375 \cdot c, -0.5\right)}{b}} \]

9. Final simplification 90.3%

   \[\leadsto c \cdot \frac{\mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot -0.375, -0.5\right)}{b} \]
10. Add Preprocessing

Alternative 9: 80.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 79.5

      \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

5. Applied rewrites 79.5%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))