Quadratic roots, wide range

Percentage Accurate: 17.7% → 97.8%

Time: 14.8s

Alternatives: 8

Speedup: 29.0×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 17.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (a * ((pow(c, 4.0) * ((-5.0 * (pow(a, 2.0) / pow(b, 7.0))) + (-2.0 * (a / (c * pow(b, 5.0)))))) - ((c * c) / pow(b, 3.0)))) - (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 = (a * (((c ** 4.0d0) * (((-5.0d0) * ((a ** 2.0d0) / (b ** 7.0d0))) + ((-2.0d0) * (a / (c * (b ** 5.0d0)))))) - ((c * c) / (b ** 3.0d0)))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 17.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 17.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 97.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

6. Taylor expanded in c around 0 97.7%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\left(20 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)}\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ 97.7%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\frac{20 \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right) \]

8. Simplified 97.7%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\frac{20 \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right) \]

9. Taylor expanded in c around inf 97.7%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + \color{blue}{{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5} \cdot c}\right)}\right) \]
10. Step-by-step derivation

    1. unpow2 97.7%

       \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5} \cdot c}\right)\right) \]

11. Applied egg-rr 97.7%

    \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5} \cdot c}\right)\right) \]

12. Final simplification 97.7%

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

Alternative 2: 97.0% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   c
   (+
    (* 2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
    (/ (* a (pow c 2.0)) (pow b 2.0))))
  (- b)))

C

double code(double a, double b, double c) {
	return (c + ((2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((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 + ((2.0d0 * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 4.0d0))) + ((a * (c ** 2.0d0)) / (b ** 2.0d0)))) / -b
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(c + Float64(Float64(2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / Float64(-b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 17.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 17.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 96.7%

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

6. Taylor expanded in b around -inf 97.0%

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

7. Final simplification 97.0%

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

Alternative 3: 96.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 17.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 17.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 96.7%

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

6. Taylor expanded in b around inf 96.7%

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

   1. mul-1-neg 96.7%

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

   2. unsub-neg 96.7%

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

   3. associate-/l* 96.7%

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

   4. unpow2 96.7%

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

   5. unpow2 96.7%

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

   6. times-frac 96.7%

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

   7. unpow2 96.7%

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

8. Simplified 96.7%

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

9. Final simplification 96.7%

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

Alternative 4: 97.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 5.0))) - ((c * c) / pow(b, 3.0)))) - (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 = (a * (((-2.0d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0))) - ((c * c) / (b ** 3.0d0)))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 17.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 17.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 97.0%

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

   1. unpow2 97.7%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5} \cdot c}\right)\right) \]

7. Applied egg-rr 97.0%

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

8. Final simplification 97.0%

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

Alternative 5: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (* a (/ c (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * (c / 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 = c * (((-1.0d0) / b) - (a * (c / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * (c / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - (a * (c / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(a * Float64(c / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - (a * (c / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 17.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 17.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 95.8%

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
6. Step-by-step derivation

   1. mul-1-neg 95.8%

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

   2. unsub-neg 95.8%

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

   3. mul-1-neg 95.8%

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

   4. associate-/l* 95.8%

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

7. Simplified 95.8%

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

8. Taylor expanded in c around 0 95.5%

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

   1. sub-neg 95.5%

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

   2. distribute-neg-frac 95.5%

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

   3. metadata-eval 95.5%

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

   4. +-commutative 95.5%

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

   5. mul-1-neg 95.5%

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

   6. unsub-neg 95.5%

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

   7. associate-/l* 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

    \[\leadsto c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \]
12. Add Preprocessing

Alternative 6: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 17.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 17.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 95.8%

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
6. Step-by-step derivation

   1. mul-1-neg 95.8%

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

   2. unsub-neg 95.8%

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

   3. mul-1-neg 95.8%

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

   4. associate-/l* 95.8%

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

7. Simplified 95.8%

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

   1. unpow2 97.7%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5} \cdot c}\right)\right) \]

9. Applied egg-rr 95.8%

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

10. Taylor expanded in c around 0 95.8%

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

12. Simplified 95.8%

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

13. Final simplification 95.8%

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

Alternative 7: 90.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 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 / -b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 17.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 17.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 90.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ 90.8%

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

   2. mul-1-neg 90.8%

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

7. Simplified 90.8%

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

8. Final simplification 90.8%

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

Alternative 8: 3.3% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 17.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 17.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 17.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 90.4%

   \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. expm1-log1p-u 80.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2}\right)\right)} \]

   2. expm1-undefine 20.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2}\right)} - 1} \]

   3. associate-/l* 20.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-2 \cdot \frac{\frac{a \cdot c}{b}}{a \cdot 2}}\right)} - 1 \]

   4. associate-/l* 20.0%

      \[\leadsto e^{\mathsf{log1p}\left(-2 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{a \cdot 2}\right)} - 1 \]

7. Applied egg-rr 20.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-2 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 2}\right)} - 1} \]
8. Step-by-step derivation

   1. sub-neg 20.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-2 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 2}\right)} + \left(-1\right)} \]

   2. metadata-eval 20.0%

      \[\leadsto e^{\mathsf{log1p}\left(-2 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 2}\right)} + \color{blue}{-1} \]

   3. +-commutative 20.0%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(-2 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 2}\right)}} \]

   4. log1p-undefine 20.0%

      \[\leadsto -1 + e^{\color{blue}{\log \left(1 + -2 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 2}\right)}} \]

   5. rem-exp-log 29.8%

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

   6. associate-*r/ 29.8%

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

   7. *-commutative 29.8%

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

   8. times-frac 29.8%

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

   9. metadata-eval 29.8%

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

   10. associate-*r/ 29.7%

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

   11. associate-/l/ 29.7%

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

   12. *-commutative 29.7%

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

   13. associate-*l/ 29.7%

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

   14. neg-mul-1 29.7%

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

   15. unsub-neg 29.7%

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

   16. associate-*l/ 29.7%

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

   17. *-commutative 29.7%

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

   18. associate-/l/ 29.7%

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

   19. associate-*r/ 29.8%

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

9. Simplified 29.8%

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

10. Taylor expanded in c around 0 3.3%

    \[\leadsto -1 + \color{blue}{1} \]

11. Final simplification 3.3%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))