Cubic critical, wide range

Percentage Accurate: 17.7% → 97.7%

Time: 16.9s

Alternatives: 9

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: 17.7% 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: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
   (+
    (* c -0.5)
    (+
     (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))
     (/ (* (pow (* a c) 4.0) -1.0546875) (* a (pow b 6.0))))))
  b))

C

double code(double a, double b, double c) {
	return ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))) + ((pow((a * c), 4.0) * -1.0546875) / (a * pow(b, 6.0)))))) / 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 = (((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 4.0d0))) + ((c * (-0.5d0)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 2.0d0))) + ((((a * c) ** 4.0d0) * (-1.0546875d0)) / (a * (b ** 6.0d0)))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 2.0))) + ((Math.pow((a * c), 4.0) * -1.0546875) / (a * Math.pow(b, 6.0)))))) / b;
}

Python

def code(a, b, c):
	return ((-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 2.0))) + ((math.pow((a * c), 4.0) * -1.0546875) / (a * math.pow(b, 6.0)))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) + Float64(Float64((Float64(a * c) ^ 4.0) * -1.0546875) / Float64(a * (b ^ 6.0)))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + ((c * -0.5) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 2.0))) + ((((a * c) ^ 4.0) * -1.0546875) / (a * (b ^ 6.0)))))) / b;
end

Wolfram

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

Derivation

1. Initial program 18.5%

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

3. Taylor expanded in b around inf 97.0%

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

   1. associate-*r/ 97.0%

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

   2. distribute-rgt-out 97.0%

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

   3. pow-prod-down 97.0%

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

   4. metadata-eval 97.0%

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

5. Applied egg-rr 97.0%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]
6. Step-by-step derivation

   1. *-commutative 97.0%

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

   2. associate-*l* 97.0%

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

   3. metadata-eval 97.0%

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

7. Simplified 97.0%

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

8. Final simplification 97.0%

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

Alternative 2: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{\left(a \cdot -1.0546875\right) \cdot {c}^{4}}{{b}^{7}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (*
   a
   (+
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
    (*
     a
     (+
      (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
      (/ (* (* a -1.0546875) (pow c 4.0)) (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (((a * -1.0546875) * pow(c, 4.0)) / pow(b, 7.0))))));
}

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 * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * (((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + (((a * (-1.0546875d0)) * (c ** 4.0d0)) / (b ** 7.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * ((-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (((a * -1.0546875) * Math.pow(c, 4.0)) / Math.pow(b, 7.0))))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * ((-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (((a * -1.0546875) * math.pow(c, 4.0)) / math.pow(b, 7.0))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(Float64(Float64(a * -1.0546875) * (c ^ 4.0)) / (b ^ 7.0)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((-0.5625 * ((c ^ 3.0) / (b ^ 5.0))) + (((a * -1.0546875) * (c ^ 4.0)) / (b ^ 7.0))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * -1.0546875), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.5%

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

3. Taylor expanded in a around 0 97.0%

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

4. Taylor expanded in c around 0 97.0%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
5. Step-by-step derivation

   1. associate-*r/ 97.0%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{\frac{-1.0546875 \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right) \]

   2. associate-*r* 97.0%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{\color{blue}{\left(-1.0546875 \cdot a\right) \cdot {c}^{4}}}{{b}^{7}}\right)\right) \]

6. Simplified 97.0%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{\frac{\left(-1.0546875 \cdot a\right) \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]

7. Final simplification 97.0%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{\left(a \cdot -1.0546875\right) \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
8. Add Preprocessing

Alternative 3: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

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 * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + ((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

3. Taylor expanded in a around 0 96.2%

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

4. Final simplification 96.2%

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

Alternative 4: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.5625 (/ (* (pow a 2.0) c) (pow b 5.0)))
     (* -0.375 (/ a (pow b 3.0)))))
   (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-0.5625 * ((pow(a, 2.0) * c) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / 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 * ((c * (((-0.5625d0) * (((a ** 2.0d0) * c) / (b ** 5.0d0))) + ((-0.375d0) * (a / (b ** 3.0d0))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-0.5625 * ((Math.pow(a, 2.0) * c) / Math.pow(b, 5.0))) + (-0.375 * (a / Math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((c * ((-0.5625 * ((math.pow(a, 2.0) * c) / math.pow(b, 5.0))) + (-0.375 * (a / math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-0.5625 * (((a ^ 2.0) * c) / (b ^ 5.0))) + (-0.375 * (a / (b ^ 3.0))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.5%

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

3. Taylor expanded in c around 0 95.9%

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

4. Final simplification 95.9%

   \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right) \]
5. Add Preprocessing

Alternative 5: 95.4% 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 18.5%

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

3. Taylor expanded in b around inf 94.3%

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

4. Final simplification 94.3%

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

Alternative 6: 95.4% 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 18.5%

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

3. Taylor expanded in a around 0 94.3%

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

Alternative 7: 95.1% accurate, 1.0× speedup

Math

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

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(Float64(-0.375 * Float64(a * 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[(N[(-0.375 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.5%

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

3. Taylor expanded in a around 0 93.8%

   \[\leadsto \frac{\color{blue}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]

4. Taylor expanded in c around 0 94.0%

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

   1. associate-*r/ 94.0%

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

   2. associate-*r/ 94.0%

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

   3. metadata-eval 94.0%

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

6. Simplified 94.0%

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

Alternative 8: 90.5% 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 18.5%

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

3. Taylor expanded in b around inf 89.7%

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

   1. associate-*r/ 89.7%

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

   2. *-commutative 89.7%

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

5. Simplified 89.7%

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

Alternative 9: 90.2% 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 18.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

   1. /-rgt-identity 18.5%

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

   2. metadata-eval 18.5%

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

3. Simplified 18.5%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 12.4%

   \[\leadsto \frac{\color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{3 \cdot a} \]
6. Step-by-step derivation

   1. associate-/l* 12.4%

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

7. Simplified 12.4%

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

8. Taylor expanded in b around 0 89.7%

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

   1. associate-*r/ 89.7%

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

   2. *-commutative 89.7%

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

   3. associate-/l* 89.3%

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

10. Simplified 89.3%

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

Reproduce

herbie shell --seed 2024094 
(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)))