Quadratic roots, medium range

Percentage Accurate: 31.3% → 95.6%

Time: 15.9s

Alternatives: 10

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.3% 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: 95.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.7%

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

   1. *-commutative 36.7%

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

3. Simplified 36.8%

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

5. Taylor expanded in a around 0 93.9%

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

6. Taylor expanded in c around inf 93.9%

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

7. Taylor expanded in c around 0 93.9%

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

8. Taylor expanded in a around 0 93.9%

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

9. Final simplification 93.9%

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

Alternative 2: 94.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.7%

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

   1. *-commutative 36.7%

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

3. Simplified 36.8%

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

5. Taylor expanded in a around 0 93.9%

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

6. Taylor expanded in c around inf 93.9%

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

7. Taylor expanded in c around 0 93.9%

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

8. Taylor expanded in b around inf 91.8%

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

10. Simplified 91.8%

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

Alternative 3: 94.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.7%

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

   1. *-commutative 36.7%

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

3. Simplified 36.8%

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

5. Taylor expanded in a around 0 93.9%

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

6. Taylor expanded in c around inf 93.9%

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

7. Taylor expanded in c around 0 91.8%

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

   1. associate-*r/ 91.8%

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

   2. *-commutative 91.8%

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

   3. *-commutative 91.8%

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

9. Simplified 91.8%

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

10. Final simplification 91.8%

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

Alternative 4: 93.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (/ (- (* -2.0 (/ (pow (* c a) 2.0) (pow 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((c * a), 2.0) / pow(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) * (((c * a) ** 2.0d0) / (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((c * a), 2.0) / Math.pow(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((c * a), 2.0) / math.pow(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((Float64(c * a) ^ 2.0) / (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 * (((c * a) ^ 2.0) / (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[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[b, 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 36.7%

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

   1. *-commutative 36.7%

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

3. Simplified 36.8%

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

5. Taylor expanded in c around 0 91.6%

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

6. Taylor expanded in b around inf 91.6%

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

   1. mul-1-neg 91.6%

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

   2. unsub-neg 91.6%

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

   3. unpow2 91.6%

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

   4. unpow2 91.6%

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

   5. swap-sqr 91.6%

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

   6. unpow2 91.6%

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

8. Simplified 91.6%

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

9. Final simplification 91.6%

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

Alternative 5: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.5e-5)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (+ c (* a (pow (/ c (- b)) 2.0))) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.5e-5) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c + (a * pow((c / -b), 2.0))) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.5e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c + Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.5e-5], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c + N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.5000000000000005e-5

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

   3. Simplified 81.6%

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

   if 9.5000000000000005e-5 < b

   1. Initial program 32.5%

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

      1. *-commutative 32.5%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in b around inf 90.3%

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

      1. mul-1-neg 90.3%

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

      2. unsub-neg 90.3%

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

      3. mul-1-neg 90.3%

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

      4. associate-/l* 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in a around 0 90.3%

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

      1. associate-/l* 90.3%

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

      2. unpow2 90.3%

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

      3. unpow2 90.3%

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

      4. times-frac 90.3%

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

      5. sqr-neg 90.3%

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

      6. unpow1 90.3%

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

      7. pow-plus 90.3%

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

      8. distribute-neg-frac2 90.3%

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

      9. metadata-eval 90.3%

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

   10. Simplified 90.3%

       \[\leadsto \frac{\left(-c\right) - \color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2}}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.5e-5)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ (+ c (* a (pow (/ c (- b)) 2.0))) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.5e-5) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c + (a * pow((c / -b), 2.0))) / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 9.5d-5) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (c + (a * ((c / -b) ** 2.0d0))) / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.5e-5) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c + (a * Math.pow((c / -b), 2.0))) / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9.5e-5:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (c + (a * math.pow((c / -b), 2.0))) / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.5e-5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c + Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9.5e-5)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (c + (a * ((c / -b) ^ 2.0))) / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.5e-5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c + N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.5000000000000005e-5

   1. Initial program 81.4%

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

   if 9.5000000000000005e-5 < b

   1. Initial program 32.5%

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

      1. *-commutative 32.5%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in b around inf 90.3%

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

      1. mul-1-neg 90.3%

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

      2. unsub-neg 90.3%

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

      3. mul-1-neg 90.3%

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

      4. associate-/l* 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in a around 0 90.3%

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

      1. associate-/l* 90.3%

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

      2. unpow2 90.3%

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

      3. unpow2 90.3%

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

      4. times-frac 90.3%

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

      5. sqr-neg 90.3%

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

      6. unpow1 90.3%

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

      7. pow-plus 90.3%

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

      8. distribute-neg-frac2 90.3%

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

      9. metadata-eval 90.3%

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

   10. Simplified 90.3%

       \[\leadsto \frac{\left(-c\right) - \color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2}}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% 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(c / Float64(-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 36.7%

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

   1. *-commutative 36.7%

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

3. Simplified 36.8%

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

5. Taylor expanded in b around inf 87.5%

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

   1. mul-1-neg 87.5%

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

   2. unsub-neg 87.5%

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

   3. mul-1-neg 87.5%

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

   4. associate-/l* 87.5%

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

7. Simplified 87.5%

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

8. Taylor expanded in a around 0 87.5%

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

   1. associate-/l* 87.5%

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

   2. unpow2 87.5%

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

   3. unpow2 87.5%

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

   4. times-frac 87.5%

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

   5. sqr-neg 87.5%

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

   6. unpow1 87.5%

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

   7. pow-plus 87.5%

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

   8. distribute-neg-frac2 87.5%

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

   9. metadata-eval 87.5%

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

10. Simplified 87.5%

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

11. Final simplification 87.5%

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

Alternative 8: 90.7% accurate, 1.0× speedup

Math

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

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 36.7%

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

   1. *-commutative 36.7%

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

3. Simplified 36.8%

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

5. Taylor expanded in c around 0 91.6%

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

6. Taylor expanded in c around 0 87.3%

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

   1. associate-*r/ 87.3%

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

   2. associate-*r* 87.3%

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

   3. neg-mul-1 87.3%

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

8. Simplified 87.3%

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

   1. add-cbrt-cube 86.7%

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

   2. pow1/3 84.4%

      \[\leadsto c \cdot \left(\frac{\left(-a\right) \cdot c}{{b}^{3}} - \color{blue}{{\left(\left(\frac{1}{b} \cdot \frac{1}{b}\right) \cdot \frac{1}{b}\right)}^{0.3333333333333333}}\right) \]

   3. pow3 84.4%

      \[\leadsto c \cdot \left(\frac{\left(-a\right) \cdot c}{{b}^{3}} - {\color{blue}{\left({\left(\frac{1}{b}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

   4. inv-pow 84.4%

      \[\leadsto c \cdot \left(\frac{\left(-a\right) \cdot c}{{b}^{3}} - {\left({\color{blue}{\left({b}^{-1}\right)}}^{3}\right)}^{0.3333333333333333}\right) \]

   5. pow-pow 84.4%

      \[\leadsto c \cdot \left(\frac{\left(-a\right) \cdot c}{{b}^{3}} - {\color{blue}{\left({b}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333}\right) \]

   6. metadata-eval 84.4%

      \[\leadsto c \cdot \left(\frac{\left(-a\right) \cdot c}{{b}^{3}} - {\left({b}^{\color{blue}{-3}}\right)}^{0.3333333333333333}\right) \]

10. Applied egg-rr 84.4%

    \[\leadsto c \cdot \left(\frac{\left(-a\right) \cdot c}{{b}^{3}} - \color{blue}{{\left({b}^{-3}\right)}^{0.3333333333333333}}\right) \]
11. Step-by-step derivation

    1. pow1 84.4%

       \[\leadsto \color{blue}{{\left(c \cdot \left(\frac{\left(-a\right) \cdot c}{{b}^{3}} - {\left({b}^{-3}\right)}^{0.3333333333333333}\right)\right)}^{1}} \]

    2. associate-/l* 84.4%

       \[\leadsto {\left(c \cdot \left(\color{blue}{\left(-a\right) \cdot \frac{c}{{b}^{3}}} - {\left({b}^{-3}\right)}^{0.3333333333333333}\right)\right)}^{1} \]

    3. div-inv 84.4%

       \[\leadsto {\left(c \cdot \left(\left(-a\right) \cdot \color{blue}{\left(c \cdot \frac{1}{{b}^{3}}\right)} - {\left({b}^{-3}\right)}^{0.3333333333333333}\right)\right)}^{1} \]

    4. pow-flip 84.4%

       \[\leadsto {\left(c \cdot \left(\left(-a\right) \cdot \left(c \cdot \color{blue}{{b}^{\left(-3\right)}}\right) - {\left({b}^{-3}\right)}^{0.3333333333333333}\right)\right)}^{1} \]

    5. metadata-eval 84.4%

       \[\leadsto {\left(c \cdot \left(\left(-a\right) \cdot \left(c \cdot {b}^{\color{blue}{-3}}\right) - {\left({b}^{-3}\right)}^{0.3333333333333333}\right)\right)}^{1} \]

    6. pow-pow 87.3%

       \[\leadsto {\left(c \cdot \left(\left(-a\right) \cdot \left(c \cdot {b}^{-3}\right) - \color{blue}{{b}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)\right)}^{1} \]

    7. metadata-eval 87.3%

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

    8. inv-pow 87.3%

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

12. Applied egg-rr 87.3%

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

    1. unpow1 87.3%

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

    2. sub-neg 87.3%

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

    3. +-commutative 87.3%

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

    4. distribute-lft-neg-out 87.3%

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

    5. unsub-neg 87.3%

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

    6. distribute-neg-frac 87.3%

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

    7. metadata-eval 87.3%

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

14. Simplified 87.3%

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

Alternative 9: 81.5% 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 36.7%

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

   1. *-commutative 36.7%

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

3. Simplified 36.8%

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

5. Taylor expanded in b around inf 77.1%

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

   1. associate-*r/ 77.1%

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

   2. mul-1-neg 77.1%

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

7. Simplified 77.1%

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

8. Final simplification 77.1%

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

Alternative 10: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 36.7%

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

   1. *-commutative 36.7%

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

3. Simplified 36.7%

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

   1. add-cbrt-cube 36.4%

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

   2. pow1/3 36.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 36.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 36.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 36.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 36.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 36.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 36.2%

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

8. Simplified 36.2%

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

   1. log1p-expm1-u 30.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\right)\right)} \]

   2. log1p-undefine 26.6%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\right)\right)} \]

   3. neg-mul-1 26.6%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\right)\right) \]

   4. fma-define 26.6%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2}\right)\right) \]

   5. pow1/3 26.5%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2}\right)\right) \]

   6. pow-pow 26.8%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2}\right)\right) \]

   7. metadata-eval 26.8%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2}\right)\right) \]

   8. *-commutative 26.8%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}{a \cdot 2}\right)\right) \]

   9. *-commutative 26.8%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}{a \cdot 2}\right)\right) \]

10. Applied egg-rr 26.8%

    \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}{a \cdot 2}\right)\right)} \]

11. Taylor expanded in c around 0 3.2%

    \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
12. Step-by-step derivation

    1. associate-*r/ 3.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b + -1 \cdot b\right)}{a}} \]

    2. distribute-rgt1-in 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

       \[\leadsto \frac{\color{blue}{0}}{a} \]

13. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024169 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))