Cubic critical, narrow range

Percentage Accurate: 55.2% → 91.0%

Time: 14.7s

Alternatives: 10

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.2% 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: 91.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot t\_0\right)\\ t_2 := c \cdot \left(c \cdot c\right)\\ \frac{\frac{b \cdot \frac{c \cdot \left(t\_2 \cdot \left(\left(\left(a \cdot a\right) \cdot 1.0546875\right) \cdot \left(0 - a\right)\right)\right)}{b} - t\_1 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot \frac{0.375}{b}\right)}{b \cdot t\_1} - \left(c \cdot 0.5 + \frac{\left(a \cdot a\right) \cdot 0.5625}{t\_0} \cdot \frac{t\_2}{b}\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b (* b t_0))) (t_2 (* c (* c c))))
   (/
    (-
     (/
      (-
       (* b (/ (* c (* t_2 (* (* (* a a) 1.0546875) (- 0.0 a)))) b))
       (* t_1 (* (* c (* c a)) (/ 0.375 b))))
      (* b t_1))
     (+ (* c 0.5) (* (/ (* (* a a) 0.5625) t_0) (/ t_2 b))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * (b * t_0);
	double t_2 = c * (c * c);
	return ((((b * ((c * (t_2 * (((a * a) * 1.0546875) * (0.0 - a)))) / b)) - (t_1 * ((c * (c * a)) * (0.375 / b)))) / (b * t_1)) - ((c * 0.5) + ((((a * a) * 0.5625) / t_0) * (t_2 / b)))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = b * (b * b)
    t_1 = b * (b * t_0)
    t_2 = c * (c * c)
    code = ((((b * ((c * (t_2 * (((a * a) * 1.0546875d0) * (0.0d0 - a)))) / b)) - (t_1 * ((c * (c * a)) * (0.375d0 / b)))) / (b * t_1)) - ((c * 0.5d0) + ((((a * a) * 0.5625d0) / t_0) * (t_2 / b)))) / b
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * (b * t_0);
	double t_2 = c * (c * c);
	return ((((b * ((c * (t_2 * (((a * a) * 1.0546875) * (0.0 - a)))) / b)) - (t_1 * ((c * (c * a)) * (0.375 / b)))) / (b * t_1)) - ((c * 0.5) + ((((a * a) * 0.5625) / t_0) * (t_2 / b)))) / b;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = b * (b * t_0)
	t_2 = c * (c * c)
	return ((((b * ((c * (t_2 * (((a * a) * 1.0546875) * (0.0 - a)))) / b)) - (t_1 * ((c * (c * a)) * (0.375 / b)))) / (b * t_1)) - ((c * 0.5) + ((((a * a) * 0.5625) / t_0) * (t_2 / b)))) / b

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(b * Float64(b * t_0))
	t_2 = Float64(c * Float64(c * c))
	return Float64(Float64(Float64(Float64(Float64(b * Float64(Float64(c * Float64(t_2 * Float64(Float64(Float64(a * a) * 1.0546875) * Float64(0.0 - a)))) / b)) - Float64(t_1 * Float64(Float64(c * Float64(c * a)) * Float64(0.375 / b)))) / Float64(b * t_1)) - Float64(Float64(c * 0.5) + Float64(Float64(Float64(Float64(a * a) * 0.5625) / t_0) * Float64(t_2 / b)))) / b)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = b * (b * t_0);
	t_2 = c * (c * c);
	tmp = ((((b * ((c * (t_2 * (((a * a) * 1.0546875) * (0.0 - a)))) / b)) - (t_1 * ((c * (c * a)) * (0.375 / b)))) / (b * t_1)) - ((c * 0.5) + ((((a * a) * 0.5625) / t_0) * (t_2 / b)))) / 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]}, Block[{t$95$2 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(b * N[(N[(c * N[(t$95$2 * N[(N[(N[(a * a), $MachinePrecision] * 1.0546875), $MachinePrecision] * N[(0.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] * N[(0.375 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(c * 0.5), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * 0.5625), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(t$95$2 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Initial program 53.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Simplified 92.0%

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

6. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(\frac{1}{2} \cdot c + \left(\frac{9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}}\right)\right)}{b}} \]

8. Simplified 92.0%

   \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b} + c \cdot 0.5\right) + \frac{0.5625 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right) + \frac{1.0546875 \cdot \left({c}^{4} \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{6}}}{0 - b}} \]

10. Applied egg-rr 92.0%

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot 1.0546875\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} + \frac{0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right) + \left(c \cdot 0.5 + \frac{0.5625 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)} \cdot \frac{c \cdot \left(c \cdot c\right)}{b}\right)}}{0 - b} \]

12. Applied egg-rr 92.0%

    \[\leadsto \frac{\color{blue}{\frac{\frac{c \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot \left(\left(a \cdot a\right) \cdot 1.0546875\right)\right)\right)}{b} \cdot b + \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot \frac{0.375}{b}\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}} + \left(c \cdot 0.5 + \frac{0.5625 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)} \cdot \frac{c \cdot \left(c \cdot c\right)}{b}\right)}{0 - b} \]

13. Final simplification 92.0%

    \[\leadsto \frac{\frac{b \cdot \frac{c \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(\left(\left(a \cdot a\right) \cdot 1.0546875\right) \cdot \left(0 - a\right)\right)\right)}{b} - \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot \frac{0.375}{b}\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} - \left(c \cdot 0.5 + \frac{\left(a \cdot a\right) \cdot 0.5625}{b \cdot \left(b \cdot b\right)} \cdot \frac{c \cdot \left(c \cdot c\right)}{b}\right)}{b} \]
14. Add Preprocessing

Alternative 2: 91.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot c\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ \frac{\left(c \cdot 0.5 + \frac{\left(a \cdot a\right) \cdot 0.5625}{t\_1} \cdot \frac{t\_0}{b}\right) + \left(\frac{\left(c \cdot t\_0\right) \cdot \left(1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot t\_1\right)\right)} + \frac{0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)}{0 - b} \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = b * (b * b);
	return (((c * 0.5) + ((((a * a) * 0.5625) / t_1) * (t_0 / b))) + ((((c * t_0) * (1.0546875 * (a * (a * a)))) / (b * (b * (b * t_1)))) + ((0.375 * ((c * c) * a)) / (b * b)))) / (0.0 - b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    t_0 = c * (c * c)
    t_1 = b * (b * b)
    code = (((c * 0.5d0) + ((((a * a) * 0.5625d0) / t_1) * (t_0 / b))) + ((((c * t_0) * (1.0546875d0 * (a * (a * a)))) / (b * (b * (b * t_1)))) + ((0.375d0 * ((c * c) * a)) / (b * b)))) / (0.0d0 - b)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = b * (b * b);
	return (((c * 0.5) + ((((a * a) * 0.5625) / t_1) * (t_0 / b))) + ((((c * t_0) * (1.0546875 * (a * (a * a)))) / (b * (b * (b * t_1)))) + ((0.375 * ((c * c) * a)) / (b * b)))) / (0.0 - b);
}

Python

def code(a, b, c):
	t_0 = c * (c * c)
	t_1 = b * (b * b)
	return (((c * 0.5) + ((((a * a) * 0.5625) / t_1) * (t_0 / b))) + ((((c * t_0) * (1.0546875 * (a * (a * a)))) / (b * (b * (b * t_1)))) + ((0.375 * ((c * c) * a)) / (b * b)))) / (0.0 - b)

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(c * c))
	t_1 = Float64(b * Float64(b * b))
	return Float64(Float64(Float64(Float64(c * 0.5) + Float64(Float64(Float64(Float64(a * a) * 0.5625) / t_1) * Float64(t_0 / b))) + Float64(Float64(Float64(Float64(c * t_0) * Float64(1.0546875 * Float64(a * Float64(a * a)))) / Float64(b * Float64(b * Float64(b * t_1)))) + Float64(Float64(0.375 * Float64(Float64(c * c) * a)) / Float64(b * b)))) / Float64(0.0 - b))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (c * c);
	t_1 = b * (b * b);
	tmp = (((c * 0.5) + ((((a * a) * 0.5625) / t_1) * (t_0 / b))) + ((((c * t_0) * (1.0546875 * (a * (a * a)))) / (b * (b * (b * t_1)))) + ((0.375 * ((c * c) * a)) / (b * b)))) / (0.0 - b);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(c * 0.5), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * 0.5625), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * t$95$0), $MachinePrecision] * N[(1.0546875 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.375 * N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 53.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Simplified 92.0%

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

6. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(\frac{1}{2} \cdot c + \left(\frac{9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}}\right)\right)}{b}} \]

8. Simplified 92.0%

   \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b} + c \cdot 0.5\right) + \frac{0.5625 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right) + \frac{1.0546875 \cdot \left({c}^{4} \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{6}}}{0 - b}} \]

10. Applied egg-rr 92.0%

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot 1.0546875\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} + \frac{0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right) + \left(c \cdot 0.5 + \frac{0.5625 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)} \cdot \frac{c \cdot \left(c \cdot c\right)}{b}\right)}}{0 - b} \]

11. Final simplification 92.0%

    \[\leadsto \frac{\left(c \cdot 0.5 + \frac{\left(a \cdot a\right) \cdot 0.5625}{b \cdot \left(b \cdot b\right)} \cdot \frac{c \cdot \left(c \cdot c\right)}{b}\right) + \left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} + \frac{0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)}{0 - b} \]
12. Add Preprocessing

Alternative 3: 91.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := c \cdot \left(c \cdot c\right)\\ \frac{\left(c \cdot 0.5 + \frac{0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right) + \left(\frac{\left(a \cdot a\right) \cdot 0.5625}{t\_0} \cdot \frac{t\_1}{b} + \frac{\left(c \cdot t\_1\right) \cdot \left(1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot t\_0\right)\right)}\right)}{0 - b} \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = c * (c * c);
	return (((c * 0.5) + ((0.375 * ((c * c) * a)) / (b * b))) + (((((a * a) * 0.5625) / t_0) * (t_1 / b)) + (((c * t_1) * (1.0546875 * (a * (a * a)))) / (b * (b * (b * t_0)))))) / (0.0 - b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    t_0 = b * (b * b)
    t_1 = c * (c * c)
    code = (((c * 0.5d0) + ((0.375d0 * ((c * c) * a)) / (b * b))) + (((((a * a) * 0.5625d0) / t_0) * (t_1 / b)) + (((c * t_1) * (1.0546875d0 * (a * (a * a)))) / (b * (b * (b * t_0)))))) / (0.0d0 - b)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = c * (c * c);
	return (((c * 0.5) + ((0.375 * ((c * c) * a)) / (b * b))) + (((((a * a) * 0.5625) / t_0) * (t_1 / b)) + (((c * t_1) * (1.0546875 * (a * (a * a)))) / (b * (b * (b * t_0)))))) / (0.0 - b);
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = c * (c * c)
	return (((c * 0.5) + ((0.375 * ((c * c) * a)) / (b * b))) + (((((a * a) * 0.5625) / t_0) * (t_1 / b)) + (((c * t_1) * (1.0546875 * (a * (a * a)))) / (b * (b * (b * t_0)))))) / (0.0 - b)

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(c * Float64(c * c))
	return Float64(Float64(Float64(Float64(c * 0.5) + Float64(Float64(0.375 * Float64(Float64(c * c) * a)) / Float64(b * b))) + Float64(Float64(Float64(Float64(Float64(a * a) * 0.5625) / t_0) * Float64(t_1 / b)) + Float64(Float64(Float64(c * t_1) * Float64(1.0546875 * Float64(a * Float64(a * a)))) / Float64(b * Float64(b * Float64(b * t_0)))))) / Float64(0.0 - b))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = c * (c * c);
	tmp = (((c * 0.5) + ((0.375 * ((c * c) * a)) / (b * b))) + (((((a * a) * 0.5625) / t_0) * (t_1 / b)) + (((c * t_1) * (1.0546875 * (a * (a * a)))) / (b * (b * (b * t_0)))))) / (0.0 - b);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(c * 0.5), $MachinePrecision] + N[(N[(0.375 * N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 0.5625), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(t$95$1 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * t$95$1), $MachinePrecision] * N[(1.0546875 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 53.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Simplified 92.0%

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

6. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(\frac{1}{2} \cdot c + \left(\frac{9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}}\right)\right)}{b}} \]

8. Simplified 92.0%

   \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b} + c \cdot 0.5\right) + \frac{0.5625 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right) + \frac{1.0546875 \cdot \left({c}^{4} \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{6}}}{0 - b}} \]

10. Applied egg-rr 92.0%

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

11. Final simplification 92.0%

    \[\leadsto \frac{\left(c \cdot 0.5 + \frac{0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right) + \left(\frac{\left(a \cdot a\right) \cdot 0.5625}{b \cdot \left(b \cdot b\right)} \cdot \frac{c \cdot \left(c \cdot c\right)}{b} + \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right)}{0 - b} \]
12. Add Preprocessing

Alternative 4: 91.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} + a \cdot \left(a \cdot \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}{b}\right) + -0.375 \cdot \frac{\frac{c}{\frac{b \cdot b}{c}}}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (/ (* c -0.5) b)
  (*
   a
   (+
    (*
     a
     (+
      (/ (* (* c c) (* c -0.5625)) (* (* b b) (* b (* b b))))
      (/
       (/
        (* (* c (* c (* c c))) (* a -1.0546875))
        (* (* b b) (* (* b b) (* b b))))
       b)))
    (* -0.375 (/ (/ c (/ (* b b) c)) b))))))

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * ((a * ((((c * c) * (c * -0.5625)) / ((b * b) * (b * (b * b)))) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * ((b * b) * (b * b)))) / b))) + (-0.375 * ((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 = ((c * (-0.5d0)) / b) + (a * ((a * ((((c * c) * (c * (-0.5625d0))) / ((b * b) * (b * (b * b)))) + ((((c * (c * (c * c))) * (a * (-1.0546875d0))) / ((b * b) * ((b * b) * (b * b)))) / b))) + ((-0.375d0) * ((c / ((b * b) / c)) / b))))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * ((a * ((((c * c) * (c * -0.5625)) / ((b * b) * (b * (b * b)))) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * ((b * b) * (b * b)))) / b))) + (-0.375 * ((c / ((b * b) / c)) / b))));
}

Python

def code(a, b, c):
	return ((c * -0.5) / b) + (a * ((a * ((((c * c) * (c * -0.5625)) / ((b * b) * (b * (b * b)))) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * ((b * b) * (b * b)))) / b))) + (-0.375 * ((c / ((b * b) / c)) / b))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(a * Float64(Float64(Float64(Float64(c * c) * Float64(c * -0.5625)) / Float64(Float64(b * b) * Float64(b * Float64(b * b)))) + Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * -1.0546875)) / Float64(Float64(b * b) * Float64(Float64(b * b) * Float64(b * b)))) / b))) + Float64(-0.375 * Float64(Float64(c / Float64(Float64(b * b) / c)) / b)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) / b) + (a * ((a * ((((c * c) * (c * -0.5625)) / ((b * b) * (b * (b * b)))) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * ((b * b) * (b * b)))) / b))) + (-0.375 * ((c / ((b * b) / c)) / b))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(a * N[(N[(N[(N[(c * c), $MachinePrecision] * N[(c * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * -1.0546875), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(c / N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Simplified 92.0%

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

7. Applied egg-rr 92.0%

   \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \color{blue}{\left(a \cdot \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}{b}\right) + -0.375 \cdot \frac{\frac{c}{\frac{b \cdot b}{c}}}{b}\right)} \]
8. Add Preprocessing

Alternative 5: 87.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((c * (-0.5d0)) + (a * ((-0.375d0) * (c * (c / (b * b)))))) + (((-0.5625d0) * (c * (c * (c * (a * a))))) / ((b * b) * (b * b)))) / b
end function

Java

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

Python

def code(a, b, c):
	return (((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) + ((-0.5625 * (c * (c * (c * (a * a))))) / ((b * b) * (b * b)))) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) + ((-0.5625 * (c * (c * (c * (a * a))))) / ((b * b) * (b * b)))) / b;
end

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in b around inf

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

5. Simplified 88.9%

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

Alternative 6: 88.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (-0.5d0)) / b) + (a * (((((-0.5625d0) * ((c * (c * c)) * a)) / (b * b)) + ((c * c) * (-0.375d0))) / (b * (b * b))))
end function

Java

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

Python

def code(a, b, c):
	return ((c * -0.5) / b) + (a * ((((-0.5625 * ((c * (c * c)) * a)) / (b * b)) + ((c * c) * -0.375)) / (b * (b * b))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) / b) + (a * ((((-0.5625 * ((c * (c * c)) * a)) / (b * b)) + ((c * c) * -0.375)) / (b * (b * b))));
end

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Simplified 92.0%

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

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}\right)}\right)\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \left({c}^{3}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \left(c \cdot \left(c \cdot c\right)\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \left(c \cdot {c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \left(b \cdot b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left({c}^{2}\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left(c \cdot c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   17. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

   18. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

   20. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

   21. *-lowering-*.f64 88.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

8. Simplified 88.9%

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

9. Final simplification 88.9%

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

Alternative 7: 81.8% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (-0.5d0)) + (a * ((-0.375d0) * (c * (c / (b * b)))))) / b
end function

Java

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

Python

def code(a, b, c):
	return ((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) / b;
end

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right), \color{blue}{b}\right) \]

5. Simplified 82.4%

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

Alternative 8: 81.7% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-0.5d0) / b) + ((c * (a * (-0.375d0))) / (b * (b * b))))
end function

Java

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

Python

def code(a, b, c):
	return c * ((-0.5 / b) + ((c * (a * -0.375)) / (b * (b * b))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.5 / b) + ((c * (a * -0.375)) / (b * (b * b))));
end

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)}\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) + \color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right), \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)}\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{b}\right)\right), \left(\left(\color{blue}{\frac{-3}{8}} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b}\right)\right), \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

   11. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b}\right), \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \left(\left(\color{blue}{\frac{-3}{8}} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right)\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c\right)\right)\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{\color{blue}{{b}^{3}}}\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{\color{blue}{b}}^{3}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\left(\frac{-3}{8} \cdot \left(a \cdot c\right)\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

5. Simplified 82.2%

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

6. Final simplification 82.2%

   \[\leadsto c \cdot \left(\frac{-0.5}{b} + \frac{c \cdot \left(a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right) \]
7. Add Preprocessing

Alternative 9: 64.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 65.4%

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

5. Simplified 65.4%

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

Alternative 10: 64.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 65.4%

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

5. Simplified 65.4%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. /-lowering-/.f64 65.3%

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

7. Applied egg-rr 65.3%

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

8. Final simplification 65.3%

   \[\leadsto c \cdot \frac{-0.5}{b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))