Quadratic roots, wide range

Percentage Accurate: 17.7% → 97.6%

Time: 19.2s

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: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -10 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   a
   (+
    (* -2.0 (/ c b))
    (*
     a
     (+
      (* -2.0 (/ (pow c 2.0) (pow b 3.0)))
      (*
       a
       (+
        (* -4.0 (/ (pow c 3.0) (pow b 5.0)))
        (* -10.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-4.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-10.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))))))) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * (((-2.0d0) * (c / b)) + (a * (((-2.0d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * (((-4.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-10.0d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0))))))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * ((-4.0 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-10.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0))))))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * ((-4.0 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-10.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0))))))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(-2.0 * Float64(c / b)) + Float64(a * Float64(Float64(-2.0 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-4.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-10.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((-2.0 * (c / b)) + (a * ((-2.0 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((-4.0 * ((c ^ 3.0) / (b ^ 5.0))) + (-10.0 * ((a * (c ^ 4.0)) / (b ^ 7.0))))))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-4.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-10.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in a around 0 98.0%

   \[\leadsto \frac{\color{blue}{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)}}{a \cdot 2} \]

6. Taylor expanded in c around 0 98.0%

   \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-10 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right)\right)}{a \cdot 2} \]
7. Add Preprocessing

Alternative 2: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -2 \cdot \frac{c}{{b}^{5}}\right) + \frac{-1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (*
     a
     (+
      (*
       a
       (+
        (* -5.0 (/ (* a (pow c 2.0)) (pow b 7.0)))
        (* -2.0 (/ c (pow b 5.0)))))
      (/ -1.0 (pow b 3.0)))))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((c * (a * ((a * ((-5.0 * ((a * pow(c, 2.0)) / pow(b, 7.0))) + (-2.0 * (c / pow(b, 5.0))))) + (-1.0 / 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 * ((c * (a * ((a * (((-5.0d0) * ((a * (c ** 2.0d0)) / (b ** 7.0d0))) + ((-2.0d0) * (c / (b ** 5.0d0))))) + ((-1.0d0) / (b ** 3.0d0))))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * (a * ((a * ((-5.0 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 7.0))) + (-2.0 * (c / Math.pow(b, 5.0))))) + (-1.0 / Math.pow(b, 3.0))))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * (a * ((a * ((-5.0 * ((a * math.pow(c, 2.0)) / math.pow(b, 7.0))) + (-2.0 * (c / math.pow(b, 5.0))))) + (-1.0 / math.pow(b, 3.0))))) + (-1.0 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(a * Float64(Float64(-5.0 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 7.0))) + Float64(-2.0 * Float64(c / (b ^ 5.0))))) + Float64(-1.0 / (b ^ 3.0))))) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * (a * ((a * ((-5.0 * ((a * (c ^ 2.0)) / (b ^ 7.0))) + (-2.0 * (c / (b ^ 5.0))))) + (-1.0 / (b ^ 3.0))))) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(a * N[(N[(a * N[(N[(-5.0 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in c around 0 97.7%

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

7. Simplified 97.7%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-1, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{a \cdot b}\right)\right)\right) - \frac{1}{b}\right)} \]

8. Taylor expanded in a around 0 97.7%

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

9. Final simplification 97.7%

   \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -2 \cdot \frac{c}{{b}^{5}}\right) + \frac{-1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right) \]
10. Add Preprocessing

Alternative 3: 96.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (- (* -2.0 (/ (* a (pow c 3.0)) (pow b 5.0))) (/ (pow c 2.0) (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))) - (pow(c, 2.0) / 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 ** 2.0d0) / (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))) - (Math.pow(c, 2.0) / 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))) - (math.pow(c, 2.0) / 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((c ^ 2.0) / (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 ^ 2.0) / (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[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in a around 0 97.5%

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

6. Final simplification 97.5%

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

Alternative 4: 96.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in c around 0 97.0%

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

6. Taylor expanded in a around 0 97.0%

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

   1. *-commutative 97.0%

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

8. Simplified 97.0%

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

9. Taylor expanded in a around 0 97.0%

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

    1. associate-*r/ 97.0%

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

    2. associate-*r* 97.0%

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

    3. exp-to-pow 97.0%

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

    4. *-commutative 97.0%

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

    5. exp-neg 97.0%

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

    6. distribute-lft-neg-in 97.0%

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

    7. metadata-eval 97.0%

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

    8. *-commutative 97.0%

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

    9. exp-to-pow 97.0%

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

11. Simplified 97.0%

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

12. Final simplification 97.0%

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

Alternative 5: 95.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in c around 0 95.9%

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

   1. associate-*r/ 95.9%

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

   2. neg-mul-1 95.9%

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

   3. distribute-lft-neg-in 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in a around -inf 95.8%

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

   1. mul-1-neg 95.8%

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

   2. *-commutative 95.8%

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

   3. distribute-rgt-neg-in 95.8%

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

   4. +-commutative 95.8%

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

   5. *-commutative 95.8%

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

10. Simplified 95.8%

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

11. Taylor expanded in b around inf 96.3%

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

    1. distribute-lft-out 96.3%

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

    2. associate-*r/ 96.3%

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

    3. mul-1-neg 96.3%

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

    4. distribute-neg-frac2 96.3%

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

    5. +-commutative 96.3%

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

    6. associate-/l* 96.3%

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

    7. fma-define 96.3%

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

    8. unpow2 96.3%

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

    9. unpow2 96.3%

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

    10. times-frac 96.3%

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

    11. unpow2 96.3%

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

13. Simplified 96.3%

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

Alternative 6: 95.0% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in c around 0 95.9%

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

   1. associate-*r/ 95.9%

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

   2. neg-mul-1 95.9%

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

   3. distribute-lft-neg-in 95.9%

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

7. Simplified 95.9%

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

8. Final simplification 95.9%

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

Alternative 7: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-1 - \frac{a \cdot c}{{b}^{2}}}{b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in c around 0 95.9%

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

   1. associate-*r/ 95.9%

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

   2. neg-mul-1 95.9%

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

   3. distribute-lft-neg-in 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in b around -inf 95.9%

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

   1. mul-1-neg 95.9%

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

   2. *-commutative 95.9%

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

10. Simplified 95.9%

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

11. Final simplification 95.9%

    \[\leadsto c \cdot \frac{-1 - \frac{a \cdot c}{{b}^{2}}}{b} \]
12. Add Preprocessing

Alternative 8: 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 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in b around inf 91.4%

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

   1. associate-*r/ 91.4%

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

   2. mul-1-neg 91.4%

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

7. Simplified 91.4%

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

8. Final simplification 91.4%

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

Reproduce

herbie shell --seed 2024114 
(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)))