Cubic critical, wide range

Percentage Accurate: 18.1% → 97.6%

Time: 22.5s

Alternatives: 10

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

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: 18.1% 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: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {a}^{4} \cdot {c}^{4}\\ \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot t\_0 + t\_0 \cdot 5.0625}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (pow a 4.0) (pow c 4.0))))
   (/
    (+
     (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
     (+
      (+
       (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))
       (*
        -0.16666666666666666
        (/ (+ (* 1.265625 t_0) (* t_0 5.0625)) (* a (pow b 6.0)))))
      (* c -0.5)))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = pow(a, 4.0) * pow(c, 4.0);
	return ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + (((-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))) + (-0.16666666666666666 * (((1.265625 * t_0) + (t_0 * 5.0625)) / (a * pow(b, 6.0))))) + (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
    real(8) :: t_0
    t_0 = (a ** 4.0d0) * (c ** 4.0d0)
    code = (((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 4.0d0))) + ((((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 2.0d0))) + ((-0.16666666666666666d0) * (((1.265625d0 * t_0) + (t_0 * 5.0625d0)) / (a * (b ** 6.0d0))))) + (c * (-0.5d0)))) / b
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(a, 4.0) * Math.pow(c, 4.0);
	return ((-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 4.0))) + (((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 2.0))) + (-0.16666666666666666 * (((1.265625 * t_0) + (t_0 * 5.0625)) / (a * Math.pow(b, 6.0))))) + (c * -0.5))) / b;
}

Python

def code(a, b, c):
	t_0 = math.pow(a, 4.0) * math.pow(c, 4.0)
	return ((-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 4.0))) + (((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 2.0))) + (-0.16666666666666666 * (((1.265625 * t_0) + (t_0 * 5.0625)) / (a * math.pow(b, 6.0))))) + (c * -0.5))) / b

Julia

function code(a, b, c)
	t_0 = Float64((a ^ 4.0) * (c ^ 4.0))
	return Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) + Float64(-0.16666666666666666 * Float64(Float64(Float64(1.265625 * t_0) + Float64(t_0 * 5.0625)) / Float64(a * (b ^ 6.0))))) + Float64(c * -0.5))) / b)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (a ^ 4.0) * (c ^ 4.0);
	tmp = ((-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + (((-0.375 * ((a * (c ^ 2.0)) / (b ^ 2.0))) + (-0.16666666666666666 * (((1.265625 * t_0) + (t_0 * 5.0625)) / (a * (b ^ 6.0))))) + (c * -0.5))) / b;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(1.265625 * t$95$0), $MachinePrecision] + N[(t$95$0 * 5.0625), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in b around inf 97.7%

   \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Final simplification 97.7%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left({a}^{4} \cdot {c}^{4}\right) \cdot 5.0625}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{6}} + -0.5625 \cdot \frac{c}{{b}^{4}}\right) + 0.375 \cdot \frac{-1}{{b}^{2}}\right)\right) - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (*
     c
     (*
      a
      (+
       (*
        a
        (+
         (* -1.0546875 (/ (* a (pow c 2.0)) (pow b 6.0)))
         (* -0.5625 (/ c (pow b 4.0)))))
       (* 0.375 (/ -1.0 (pow b 2.0))))))
    0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * ((c * (a * ((a * ((-1.0546875 * ((a * pow(c, 2.0)) / pow(b, 6.0))) + (-0.5625 * (c / pow(b, 4.0))))) + (0.375 * (-1.0 / pow(b, 2.0)))))) - 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 * ((c * (a * ((a * (((-1.0546875d0) * ((a * (c ** 2.0d0)) / (b ** 6.0d0))) + ((-0.5625d0) * (c / (b ** 4.0d0))))) + (0.375d0 * ((-1.0d0) / (b ** 2.0d0)))))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((c * (a * ((a * ((-1.0546875 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 6.0))) + (-0.5625 * (c / Math.pow(b, 4.0))))) + (0.375 * (-1.0 / Math.pow(b, 2.0)))))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((c * (a * ((a * ((-1.0546875 * ((a * math.pow(c, 2.0)) / math.pow(b, 6.0))) + (-0.5625 * (c / math.pow(b, 4.0))))) + (0.375 * (-1.0 / math.pow(b, 2.0)))))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(a * Float64(Float64(-1.0546875 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 6.0))) + Float64(-0.5625 * Float64(c / (b ^ 4.0))))) + Float64(0.375 * Float64(-1.0 / (b ^ 2.0)))))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((c * (a * ((a * ((-1.0546875 * ((a * (c ^ 2.0)) / (b ^ 6.0))) + (-0.5625 * (c / (b ^ 4.0))))) + (0.375 * (-1.0 / (b ^ 2.0)))))) - 0.5)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(c * N[(a * N[(N[(a * N[(N[(-1.0546875 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.375 * N[(-1.0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in b around inf 97.7%

   \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Taylor expanded in c around 0 97.6%

   \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{4}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{6}}\right)\right) - 0.5\right)}}{b} \]

5. Taylor expanded in a around 0 97.6%

   \[\leadsto \frac{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{4}} + \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}}\right)\right) - 0.5\right)}{b} \]

6. Taylor expanded in a around 0 97.6%

   \[\leadsto \frac{c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{6}} + -0.5625 \cdot \frac{c}{{b}^{4}}\right) - 0.375 \cdot \frac{1}{{b}^{2}}\right)\right)} - 0.5\right)}{b} \]

7. Final simplification 97.6%

   \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{6}} + -0.5625 \cdot \frac{c}{{b}^{4}}\right) + 0.375 \cdot \frac{-1}{{b}^{2}}\right)\right) - 0.5\right)}{b} \]
8. Add Preprocessing

Alternative 3: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (*
   a
   (+
    (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * (pow(c, 2.0) / pow(b, 3.0)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + (a * (((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0))) + ((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0)))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0)))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0)))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * (c ^ 3.0)) / (b ^ 5.0))) + (-0.375 * ((c ^ 2.0) / (b ^ 3.0)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in a around 0 96.9%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
4. Add Preprocessing

Alternative 4: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - \frac{0.375}{{b}^{2}}\right)\right) - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (* c (* a (- (* -0.5625 (/ (* a c) (pow b 4.0))) (/ 0.375 (pow b 2.0)))))
    0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * ((c * (a * ((-0.5625 * ((a * c) / pow(b, 4.0))) - (0.375 / pow(b, 2.0))))) - 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 * ((c * (a * (((-0.5625d0) * ((a * c) / (b ** 4.0d0))) - (0.375d0 / (b ** 2.0d0))))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((c * (a * ((-0.5625 * ((a * c) / Math.pow(b, 4.0))) - (0.375 / Math.pow(b, 2.0))))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((c * (a * ((-0.5625 * ((a * c) / math.pow(b, 4.0))) - (0.375 / math.pow(b, 2.0))))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * c) / (b ^ 4.0))) - Float64(0.375 / (b ^ 2.0))))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((c * (a * ((-0.5625 * ((a * c) / (b ^ 4.0))) - (0.375 / (b ^ 2.0))))) - 0.5)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(c * N[(a * N[(N[(-0.5625 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in b around inf 97.7%

   \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Taylor expanded in c around 0 97.6%

   \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{4}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{6}}\right)\right) - 0.5\right)}}{b} \]

5. Taylor expanded in a around 0 97.6%

   \[\leadsto \frac{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{4}} + \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}}\right)\right) - 0.5\right)}{b} \]

6. Taylor expanded in a around 0 96.8%

   \[\leadsto \frac{c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot \frac{1}{{b}^{2}}\right)\right)} - 0.5\right)}{b} \]
7. Step-by-step derivation

   1. associate-*r/ 96.8%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - \color{blue}{\frac{0.375 \cdot 1}{{b}^{2}}}\right)\right) - 0.5\right)}{b} \]

   2. metadata-eval 96.8%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - \frac{\color{blue}{0.375}}{{b}^{2}}\right)\right) - 0.5\right)}{b} \]

8. Simplified 96.8%

   \[\leadsto \frac{c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - \frac{0.375}{{b}^{2}}\right)\right)} - 0.5\right)}{b} \]
9. Add Preprocessing

Alternative 5: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in a around 0 95.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
4. Add Preprocessing

Alternative 6: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -0.5)) b))

C

double code(double a, double b, double c) {
	return fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-0.375, Float64(a * (Float64(c / b) ^ 2.0)), Float64(c * -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in a around 0 95.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Taylor expanded in b around inf 95.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
5. Step-by-step derivation

   1. +-commutative 95.4%

      \[\leadsto \frac{\color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.5 \cdot c}}{b} \]

   2. associate-*r/ 95.4%

      \[\leadsto \frac{-0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)} + -0.5 \cdot c}{b} \]

   3. *-commutative 95.4%

      \[\leadsto \frac{-0.375 \cdot \left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right) + \color{blue}{c \cdot -0.5}}{b} \]

   4. fma-define 95.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, c \cdot -0.5\right)}}{b} \]

   5. unpow2 95.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}, c \cdot -0.5\right)}{b} \]

   6. unpow2 95.4%

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

   7. times-frac 95.4%

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

   8. unpow1 95.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right), c \cdot -0.5\right)}{b} \]

   9. pow-plus 95.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}, c \cdot -0.5\right)}{b} \]

   10. metadata-eval 95.4%

       \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}, c \cdot -0.5\right)}{b} \]

6. Simplified 95.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}} \]
7. Add Preprocessing

Alternative 7: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (- (* -0.375 (/ (* a c) (pow b 2.0))) 0.5)) b))

C

double code(double a, double b, double c) {
	return (c * ((-0.375 * ((a * c) / pow(b, 2.0))) - 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 * (((-0.375d0) * ((a * c) / (b ** 2.0d0))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((-0.375 * ((a * c) / Math.pow(b, 2.0))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((-0.375 * ((a * c) / math.pow(b, 2.0))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 2.0))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((-0.375 * ((a * c) / (b ^ 2.0))) - 0.5)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in b around inf 97.7%

   \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Taylor expanded in c around 0 95.4%

   \[\leadsto \frac{\color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}}{b} \]
5. Add Preprocessing

Alternative 8: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (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 * (((-0.375d0) * (a * (c / (b ** 3.0d0)))) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in a around 0 95.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Taylor expanded in c around 0 95.1%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
5. Step-by-step derivation

   1. associate-/l* 95.1%

      \[\leadsto c \cdot \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{3}}\right)} - 0.5 \cdot \frac{1}{b}\right) \]

   2. associate-*r/ 95.1%

      \[\leadsto c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   3. metadata-eval 95.1%

      \[\leadsto c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{\color{blue}{0.5}}{b}\right) \]

6. Simplified 95.1%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)} \]
7. Add Preprocessing

Alternative 9: 90.2% 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 17.5%

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

3. Taylor expanded in b around inf 90.5%

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

   1. associate-*r/ 90.5%

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

   2. *-commutative 90.5%

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

5. Simplified 90.5%

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

Alternative 10: 89.9% 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 17.5%

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

3. Taylor expanded in b around inf 90.5%

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

   1. associate-*r/ 90.5%

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

   2. *-commutative 90.5%

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

5. Simplified 90.5%

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

6. Taylor expanded in c around 0 90.5%

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

   1. *-commutative 90.5%

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

   2. associate-*l/ 90.5%

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

   3. associate-*r/ 90.2%

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

8. Simplified 90.2%

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

Reproduce

herbie shell --seed 2024112 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))