Cubic critical, medium range

Percentage Accurate: 32.5% → 95.4%

Time: 14.0s

Alternatives: 11

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: 32.5% 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: 95.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{b \cdot t\_1}\\ t_4 := c \cdot \left(a \cdot c\right)\\ t_5 := \left(c \cdot c\right) \cdot \left(t\_4 \cdot -1.0546875\right)\\ \mathsf{fma}\left(\frac{\frac{t\_5 \cdot \left(\left(c \cdot \left(c \cdot t\_4\right)\right) \cdot 1.0546875\right)}{t\_2 \cdot t\_2} - t\_3 \cdot t\_3}{\frac{t\_5}{t\_2} - t\_3}, a \cdot a, \mathsf{fma}\left(c \cdot c, a \cdot \frac{-0.375}{t\_0}, \frac{c \cdot -0.5}{b}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (* (* b b) (* b b)))
        (t_2 (* t_0 t_1))
        (t_3 (/ (* c (* c (* c -0.5625))) (* b t_1)))
        (t_4 (* c (* a c)))
        (t_5 (* (* c c) (* t_4 -1.0546875))))
   (fma
    (/
     (- (/ (* t_5 (* (* c (* c t_4)) 1.0546875)) (* t_2 t_2)) (* t_3 t_3))
     (- (/ t_5 t_2) t_3))
    (* a a)
    (fma (* c c) (* a (/ -0.375 t_0)) (/ (* c -0.5) b)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * (b * b);
	double t_2 = t_0 * t_1;
	double t_3 = (c * (c * (c * -0.5625))) / (b * t_1);
	double t_4 = c * (a * c);
	double t_5 = (c * c) * (t_4 * -1.0546875);
	return fma(((((t_5 * ((c * (c * t_4)) * 1.0546875)) / (t_2 * t_2)) - (t_3 * t_3)) / ((t_5 / t_2) - t_3)), (a * a), fma((c * c), (a * (-0.375 / t_0)), ((c * -0.5) / b)));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * Float64(b * b))
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(Float64(c * Float64(c * Float64(c * -0.5625))) / Float64(b * t_1))
	t_4 = Float64(c * Float64(a * c))
	t_5 = Float64(Float64(c * c) * Float64(t_4 * -1.0546875))
	return fma(Float64(Float64(Float64(Float64(t_5 * Float64(Float64(c * Float64(c * t_4)) * 1.0546875)) / Float64(t_2 * t_2)) - Float64(t_3 * t_3)) / Float64(Float64(t_5 / t_2) - t_3)), Float64(a * a), fma(Float64(c * c), Float64(a * Float64(-0.375 / t_0)), Float64(Float64(c * -0.5) / b)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * N[(c * N[(c * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(c * c), $MachinePrecision] * N[(t$95$4 * -1.0546875), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(t$95$5 * N[(N[(c * N[(c * t$95$4), $MachinePrecision]), $MachinePrecision] * 1.0546875), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$5 / t$95$2), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(a * N[(-0.375 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

7. Applied rewrites 96.3%

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

9. Applied rewrites 96.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)\right) \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \color{blue}{a \cdot a}, \mathsf{fma}\left(c \cdot c, \frac{-0.375}{b \cdot \left(b \cdot b\right)} \cdot a, \frac{c \cdot -0.5}{b}\right)\right) \]

11. Applied rewrites 96.4%

    \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot -1.0546875\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot \left(c \cdot a\right)\right)\right)\right) \cdot 1.0546875\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)} - \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} \cdot \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}{\frac{\left(c \cdot c\right) \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot -1.0546875\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} - \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}, \color{blue}{a} \cdot a, \mathsf{fma}\left(c \cdot c, \frac{-0.375}{b \cdot \left(b \cdot b\right)} \cdot a, \frac{c \cdot -0.5}{b}\right)\right) \]

12. Final simplification 96.4%

    \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot \left(a \cdot c\right)\right) \cdot -1.0546875\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot \left(a \cdot c\right)\right)\right)\right) \cdot 1.0546875\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)} - \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} \cdot \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}{\frac{\left(c \cdot c\right) \cdot \left(\left(c \cdot \left(a \cdot c\right)\right) \cdot -1.0546875\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} - \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}, a \cdot a, \mathsf{fma}\left(c \cdot c, a \cdot \frac{-0.375}{b \cdot \left(b \cdot b\right)}, \frac{c \cdot -0.5}{b}\right)\right) \]
13. Add Preprocessing

Alternative 2: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot t\_0\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{t\_1}, \frac{-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot t\_1}\right), a \cdot a, \mathsf{fma}\left(c \cdot c, a \cdot \frac{-0.375}{t\_0}, \frac{c \cdot -0.5}{b}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b (* b t_0))))
   (fma
    (fma
     c
     (* c (/ (* c -0.5625) t_1))
     (/ (* -1.0546875 (* c (* c (* a (* c c))))) (* (* b b) t_1)))
    (* a a)
    (fma (* c c) (* a (/ -0.375 t_0)) (/ (* c -0.5) b)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * (b * t_0);
	return fma(fma(c, (c * ((c * -0.5625) / t_1)), ((-1.0546875 * (c * (c * (a * (c * c))))) / ((b * b) * t_1))), (a * a), fma((c * c), (a * (-0.375 / t_0)), ((c * -0.5) / b)));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(b * Float64(b * t_0))
	return fma(fma(c, Float64(c * Float64(Float64(c * -0.5625) / t_1)), Float64(Float64(-1.0546875 * Float64(c * Float64(c * Float64(a * Float64(c * c))))) / Float64(Float64(b * b) * t_1))), Float64(a * a), fma(Float64(c * c), Float64(a * Float64(-0.375 / t_0)), Float64(Float64(c * -0.5) / b)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * N[(c * N[(N[(c * -0.5625), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[(c * N[(c * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(a * N[(-0.375 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

7. Applied rewrites 96.3%

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

9. Applied rewrites 96.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)\right) \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \color{blue}{a \cdot a}, \mathsf{fma}\left(c \cdot c, \frac{-0.375}{b \cdot \left(b \cdot b\right)} \cdot a, \frac{c \cdot -0.5}{b}\right)\right) \]

10. Final simplification 96.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), a \cdot a, \mathsf{fma}\left(c \cdot c, a \cdot \frac{-0.375}{b \cdot \left(b \cdot b\right)}, \frac{c \cdot -0.5}{b}\right)\right) \]
11. Add Preprocessing

Alternative 3: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot c\right)\\ t_1 := b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{t\_1}, \frac{-1.0546875 \cdot \left(c \cdot \left(c \cdot t\_0\right)\right)}{\left(b \cdot b\right) \cdot t\_1}\right), a \cdot a, \frac{\mathsf{fma}\left(-0.375, \frac{t\_0}{b \cdot b}, c \cdot -0.5\right)}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c c))) (t_1 (* b (* b (* b (* b b))))))
   (fma
    (fma
     c
     (* c (/ (* c -0.5625) t_1))
     (/ (* -1.0546875 (* c (* c t_0))) (* (* b b) t_1)))
    (* a a)
    (/ (fma -0.375 (/ t_0 (* b b)) (* c -0.5)) b))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * c);
	double t_1 = b * (b * (b * (b * b)));
	return fma(fma(c, (c * ((c * -0.5625) / t_1)), ((-1.0546875 * (c * (c * t_0))) / ((b * b) * t_1))), (a * a), (fma(-0.375, (t_0 / (b * b)), (c * -0.5)) / b));
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * c))
	t_1 = Float64(b * Float64(b * Float64(b * Float64(b * b))))
	return fma(fma(c, Float64(c * Float64(Float64(c * -0.5625) / t_1)), Float64(Float64(-1.0546875 * Float64(c * Float64(c * t_0))) / Float64(Float64(b * b) * t_1))), Float64(a * a), Float64(fma(-0.375, Float64(t_0 / Float64(b * b)), Float64(c * -0.5)) / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * N[(c * N[(N[(c * -0.5625), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[(c * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(-0.375 * N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

7. Applied rewrites 96.3%

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

9. Applied rewrites 96.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)\right) \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \color{blue}{a \cdot a}, \mathsf{fma}\left(c \cdot c, \frac{-0.375}{b \cdot \left(b \cdot b\right)} \cdot a, \frac{c \cdot -0.5}{b}\right)\right) \]

10. Taylor expanded in b around inf

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{c \cdot \frac{-9}{16}}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)\right) \cdot \frac{-135}{128}}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), a \cdot a, \frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\right) \]
11. Step-by-step derivation

12. Applied rewrites 96.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)\right) \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), a \cdot a, \frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\right) \]

13. Final simplification 96.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), a \cdot a, \frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\right) \]
14. Add Preprocessing

Alternative 4: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot t\_0\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{t\_1}, \frac{-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot t\_1}\right), \frac{c \cdot \left(c \cdot -0.375\right)}{t\_0}\right), a, \frac{c \cdot -0.5}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b (* b t_0))))
   (fma
    (fma
     a
     (fma
      c
      (* c (/ (* c -0.5625) t_1))
      (/ (* -1.0546875 (* c (* c (* a (* c c))))) (* (* b b) t_1)))
     (/ (* c (* c -0.375)) t_0))
    a
    (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * (b * t_0);
	return fma(fma(a, fma(c, (c * ((c * -0.5625) / t_1)), ((-1.0546875 * (c * (c * (a * (c * c))))) / ((b * b) * t_1))), ((c * (c * -0.375)) / t_0)), a, ((c * -0.5) / b));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(b * Float64(b * t_0))
	return fma(fma(a, fma(c, Float64(c * Float64(Float64(c * -0.5625) / t_1)), Float64(Float64(-1.0546875 * Float64(c * Float64(c * Float64(a * Float64(c * c))))) / Float64(Float64(b * b) * t_1))), Float64(Float64(c * Float64(c * -0.375)) / t_0)), a, Float64(Float64(c * -0.5) / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(c * N[(c * N[(N[(c * -0.5625), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[(c * N[(c * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * N[(c * -0.375), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

7. Applied rewrites 96.3%

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

9. Applied rewrites 96.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)\right) \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right), \color{blue}{a}, \frac{c \cdot -0.5}{b}\right) \]

10. Final simplification 96.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right), a, \frac{c \cdot -0.5}{b}\right) \]
11. Add Preprocessing

Alternative 5: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot t\_0\right)\\ \mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{t\_1}, \frac{-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot t\_1}\right), \frac{c \cdot \left(c \cdot -0.375\right)}{t\_0}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b (* b t_0))))
   (fma
    (/ -0.5 b)
    c
    (*
     a
     (fma
      a
      (fma
       c
       (* c (/ (* c -0.5625) t_1))
       (/ (* -1.0546875 (* c (* c (* a (* c c))))) (* (* b b) t_1)))
      (/ (* c (* c -0.375)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * (b * t_0);
	return fma((-0.5 / b), c, (a * fma(a, fma(c, (c * ((c * -0.5625) / t_1)), ((-1.0546875 * (c * (c * (a * (c * c))))) / ((b * b) * t_1))), ((c * (c * -0.375)) / t_0))));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(b * Float64(b * t_0))
	return fma(Float64(-0.5 / b), c, Float64(a * fma(a, fma(c, Float64(c * Float64(Float64(c * -0.5625) / t_1)), Float64(Float64(-1.0546875 * Float64(c * Float64(c * Float64(a * Float64(c * c))))) / Float64(Float64(b * b) * t_1))), Float64(Float64(c * Float64(c * -0.375)) / t_0))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(-0.5 / b), $MachinePrecision] * c + N[(a * N[(a * N[(c * N[(c * N[(N[(c * -0.5625), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[(c * N[(c * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * N[(c * -0.375), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

7. Applied rewrites 96.3%

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

9. Applied rewrites 96.0%

   \[\leadsto \mathsf{fma}\left(\frac{-0.5}{b}, \color{blue}{c}, a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)\right) \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right)\right) \]

10. Final simplification 96.0%

    \[\leadsto \mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(c, c \cdot \frac{c \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right)\right) \]
11. Add Preprocessing

Alternative 6: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{c \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}\right), c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   a
   (fma
    a
    (/ (* c (* c (* c -0.5625))) (* b (* b (* b b))))
    (/ (* c c) (* (* b b) -2.6666666666666665)))
   (* c -0.5))
  b))

C

double code(double a, double b, double c) {
	return fma(a, fma(a, ((c * (c * (c * -0.5625))) / (b * (b * (b * b)))), ((c * c) / ((b * b) * -2.6666666666666665))), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(a, fma(a, Float64(Float64(c * Float64(c * Float64(c * -0.5625))) / Float64(b * Float64(b * Float64(b * b)))), Float64(Float64(c * c) / Float64(Float64(b * b) * -2.6666666666666665))), Float64(c * -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(a * N[(N[(c * N[(c * N[(c * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 94.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)\right)}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 94.4%

   \[\leadsto \mathsf{fma}\left(a \cdot a, \frac{\left(c \cdot c\right) \cdot \left(c \cdot -0.5625\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \mathsf{fma}\left(a \cdot \left(c \cdot c\right), \frac{-0.375}{b \cdot b}, c \cdot -0.5\right)\right) \cdot \color{blue}{\frac{1}{b}} \]
8. Step-by-step derivation

9. Applied rewrites 94.7%

   \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{c \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}\right), c \cdot -0.5\right)}{\color{blue}{b}} \]
10. Add Preprocessing

Alternative 7: 93.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

6. Taylor expanded in b around inf

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

8. Applied rewrites 94.7%

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

Alternative 8: 90.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

6. Taylor expanded in a around 0

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

8. Applied rewrites 91.7%

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

9. Final simplification 91.7%

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

Alternative 9: 90.3% 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 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 91.7%

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

Alternative 10: 90.2% 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 31.6%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 94.7%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 91.6%

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

9. Final simplification 91.6%

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

Alternative 11: 80.5% 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 31.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 81.4

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

5. Applied rewrites 81.4%

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

Reproduce

herbie shell --seed 2024232 
(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)))