Cubic critical, wide range

Percentage Accurate: 17.8% → 97.3%

Time: 11.6s

Alternatives: 8

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: 17.8% 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: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (-
   (*
    c
    (*
     a
     (-
      (*
       a
       (+
        (* -1.0546875 (/ (* a (pow c 2.0)) (pow b 7.0)))
        (* -0.5625 (/ c (pow b 5.0)))))
      (/ 0.375 (pow b 3.0)))))
   (/ 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, 7.0))) + (-0.5625 * (c / pow(b, 5.0))))) - (0.375 / 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 * ((c * (a * ((a * (((-1.0546875d0) * ((a * (c ** 2.0d0)) / (b ** 7.0d0))) + ((-0.5625d0) * (c / (b ** 5.0d0))))) - (0.375d0 / (b ** 3.0d0))))) - (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, 7.0))) + (-0.5625 * (c / Math.pow(b, 5.0))))) - (0.375 / Math.pow(b, 3.0))))) - (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, 7.0))) + (-0.5625 * (c / math.pow(b, 5.0))))) - (0.375 / math.pow(b, 3.0))))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(a * Float64(Float64(-1.0546875 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 7.0))) + Float64(-0.5625 * Float64(c / (b ^ 5.0))))) - Float64(0.375 / (b ^ 3.0))))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * (a * ((a * ((-1.0546875 * ((a * (c ^ 2.0)) / (b ^ 7.0))) + (-0.5625 * (c / (b ^ 5.0))))) - (0.375 / (b ^ 3.0))))) - (0.5 / b));
end

Wolfram

code[a_, b_, c_] := 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, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in c around 0 96.9%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

5. Simplified 96.9%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

6. Taylor expanded in a around 0 96.9%

   \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right)} - \frac{0.5}{b}\right) \]

7. Taylor expanded in b around 0 96.9%

   \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) - \color{blue}{\frac{0.375}{{b}^{3}}}\right)\right) - \frac{0.5}{b}\right) \]

8. Final simplification 96.9%

   \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]
9. Add Preprocessing

Alternative 2: 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 15.0%

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

3. Taylor expanded in a around 0 96.7%

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

4. Final simplification 96.7%

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

Alternative 3: 96.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((c * (a * ((-0.5625 * (a * (c / pow(b, 5.0)))) - (0.375 / 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 * ((c * (a * (((-0.5625d0) * (a * (c / (b ** 5.0d0)))) - (0.375d0 / (b ** 3.0d0))))) - (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, 5.0)))) - (0.375 / Math.pow(b, 3.0))))) - (0.5 / b));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in c around 0 96.9%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

5. Simplified 96.9%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

6. Taylor expanded in a around 0 96.3%

   \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right)} - \frac{0.5}{b}\right) \]
7. Step-by-step derivation

   1. associate-/l* 96.3%

      \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{5}}\right)} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

   2. associate-*r/ 96.3%

      \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \left(a \cdot \frac{c}{{b}^{5}}\right) - \color{blue}{\frac{0.375 \cdot 1}{{b}^{3}}}\right)\right) - \frac{0.5}{b}\right) \]

   3. metadata-eval 96.3%

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

8. Simplified 96.3%

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

9. Final simplification 96.3%

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

Alternative 4: 95.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in b around inf 95.9%

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

4. Final simplification 95.9%

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

Alternative 5: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in b around inf 95.9%

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

4. Taylor expanded in c around 0 95.5%

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

   1. associate-*r/ 95.5%

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

   2. associate-*r/ 95.5%

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

   3. metadata-eval 95.5%

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

6. Simplified 95.5%

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

7. Final simplification 95.5%

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

Alternative 6: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in b around inf 95.9%

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

4. Taylor expanded in c around 0 95.5%

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

   1. associate-*r/ 95.5%

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

   2. associate-*r/ 95.5%

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

   3. metadata-eval 95.5%

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

6. Simplified 95.5%

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

7. Taylor expanded in b around inf 95.5%

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

8. Final simplification 95.5%

   \[\leadsto c \cdot \frac{-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5}{b} \]
9. Add Preprocessing

Alternative 7: 90.1% 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 15.0%

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

3. Taylor expanded in b around inf 91.5%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
4. Step-by-step derivation

   1. associate-*r/ 91.6%

      \[\leadsto \frac{\color{blue}{\frac{-1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]

   2. *-commutative 91.6%

      \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot -1.5}}{b}}{3 \cdot a} \]

5. Simplified 91.6%

   \[\leadsto \frac{\color{blue}{\frac{\left(a \cdot c\right) \cdot -1.5}{b}}}{3 \cdot a} \]

6. Taylor expanded in a around 0 92.2%

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

   1. associate-*r/ 92.2%

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

   2. *-commutative 92.2%

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

   3. associate-/l* 91.8%

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

8. Simplified 91.8%

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

9. Final simplification 91.8%

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

Alternative 8: 90.4% 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 15.0%

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

3. Taylor expanded in b around inf 92.2%

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

   1. associate-*r/ 92.2%

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

   2. *-commutative 92.2%

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

5. Simplified 92.2%

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

6. Final simplification 92.2%

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

Reproduce

herbie shell --seed 2024081 
(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)))