Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.8%

Time: 14.1s

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: 17.8% 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, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (*
    (* c c)
    (+
     (* c (* a (- (* -5.0 (* a (/ c (pow b 7.0)))) (/ 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 * ((c * c) * ((c * (a * ((-5.0 * (a * (c / pow(b, 7.0)))) - (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 * c) * ((c * (a * (((-5.0d0) * (a * (c / (b ** 7.0d0)))) - (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 * ((c * c) * ((c * (a * ((-5.0 * (a * (c / Math.pow(b, 7.0)))) - (2.0 / Math.pow(b, 5.0))))) + (-1.0 / Math.pow(b, 3.0))))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((c * c) * ((c * (a * ((-5.0 * (a * (c / math.pow(b, 7.0)))) - (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(Float64(c * c) * Float64(Float64(c * Float64(a * Float64(Float64(-5.0 * Float64(a * Float64(c / (b ^ 7.0)))) - Float64(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 * c) * ((c * (a * ((-5.0 * (a * (c / (b ^ 7.0)))) - (2.0 / (b ^ 5.0))))) + (-1.0 / (b ^ 3.0))))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(c * c), $MachinePrecision] * N[(N[(c * N[(a * N[(N[(-5.0 * N[(a * N[(c / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[Power[b, 5.0], $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 18.1%

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

   1. *-commutative 18.1%

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

3. Simplified 18.1%

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

5. Taylor expanded in a around 0 97.2%

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

   1. +-commutative 97.2%

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

   2. mul-1-neg 97.2%

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

   3. unsub-neg 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in c around 0 97.2%

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

9. Taylor expanded in a around 0 97.2%

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

    1. associate-/l* 97.2%

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

    2. associate-*r/ 97.2%

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

    3. metadata-eval 97.2%

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

11. Simplified 97.2%

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

    1. unpow2 97.2%

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

13. Applied egg-rr 97.2%

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

14. Final simplification 97.2%

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

Alternative 2: 97.0% accurate, 0.4× speedup

Math

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

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(c * Float64(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[(c * N[(a * -2.0), $MachinePrecision]), $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 18.1%

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

   1. *-commutative 18.1%

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

3. Simplified 18.1%

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

5. Taylor expanded in a around 0 97.2%

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

   1. +-commutative 97.2%

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

   2. mul-1-neg 97.2%

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

   3. unsub-neg 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in c around 0 96.6%

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

   1. associate-*r/ 96.6%

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

   2. associate-*r* 96.6%

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

10. Simplified 96.6%

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

11. Final simplification 96.6%

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

Alternative 3: 96.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

   1. *-commutative 18.1%

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

3. Simplified 18.1%

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

5. Taylor expanded in c around 0 96.3%

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

   1. unpow2 96.3%

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

7. Applied egg-rr 96.3%

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

8. Final simplification 96.3%

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

Alternative 4: 95.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

   1. *-commutative 18.1%

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

3. Simplified 18.1%

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

5. Taylor expanded in b around inf 95.3%

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

   1. mul-1-neg 95.3%

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

   2. unsub-neg 95.3%

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

   3. mul-1-neg 95.3%

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

   4. associate-/l* 95.3%

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

7. Simplified 95.3%

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

8. Final simplification 95.3%

   \[\leadsto \frac{a \cdot \frac{{c}^{2}}{-{b}^{2}} - c}{b} \]
9. 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 18.1%

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

   1. *-commutative 18.1%

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

3. Simplified 18.1%

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

5. Taylor expanded in a around 0 97.2%

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

   1. +-commutative 97.2%

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

   2. mul-1-neg 97.2%

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

   3. unsub-neg 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in b around inf 95.3%

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

   1. sub-neg 95.3%

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

   2. mul-1-neg 95.3%

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

   3. distribute-neg-out 95.3%

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

   4. remove-double-neg 95.3%

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

   5. sub-neg 95.3%

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

   6. neg-mul-1 95.3%

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

   7. distribute-neg-frac 95.3%

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

   8. distribute-neg-frac2 95.3%

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

   9. associate-/l* 95.3%

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

   10. neg-mul-1 95.3%

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

   11. fmm-def 95.3%

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

   12. unpow2 95.3%

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

   13. unpow2 95.3%

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

   14. times-frac 95.3%

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

   15. unpow2 95.3%

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

   16. remove-double-neg 95.3%

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

10. Simplified 95.3%

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

Alternative 6: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

   1. *-commutative 18.1%

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

3. Simplified 18.1%

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

5. Taylor expanded in a around 0 97.2%

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

   1. +-commutative 97.2%

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

   2. mul-1-neg 97.2%

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

   3. unsub-neg 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in c around 0 95.1%

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

   1. sub-neg 95.1%

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

   2. associate-*r/ 95.1%

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

   3. neg-mul-1 95.1%

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

   4. distribute-rgt-neg-in 95.1%

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

   5. associate-*r/ 95.1%

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

   6. +-commutative 95.1%

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

   7. associate-*r/ 95.1%

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

   8. distribute-rgt-neg-in 95.1%

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

   9. distribute-frac-neg 95.1%

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

   10. unsub-neg 95.1%

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

   11. distribute-neg-frac 95.1%

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

   12. metadata-eval 95.1%

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

   13. *-commutative 95.1%

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

   14. associate-*r/ 95.1%

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

10. Simplified 95.1%

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

Alternative 7: 94.9% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, c):
	return c * (c * (((-1.0 / c) - (a / (b * b))) / b))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

   1. *-commutative 18.1%

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

3. Simplified 18.1%

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

5. Taylor expanded in c around 0 95.1%

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

   1. associate-*r/ 95.1%

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

   2. neg-mul-1 95.1%

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

   3. distribute-rgt-neg-in 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in c around inf 94.8%

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

   1. associate-*r/ 94.8%

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

   2. mul-1-neg 94.8%

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

   3. *-commutative 94.8%

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

10. Simplified 94.8%

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

11. Taylor expanded in b around inf 94.8%

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

    1. unpow2 94.8%

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

13. Applied egg-rr 94.8%

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

14. Final simplification 94.8%

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

Alternative 8: 90.4% 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 18.1%

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

   1. *-commutative 18.1%

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

3. Simplified 18.1%

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

5. Taylor expanded in a around 0 90.3%

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

   1. associate-*r/ 90.3%

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

   2. mul-1-neg 90.3%

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

7. Simplified 90.3%

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

8. Final simplification 90.3%

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

Reproduce

herbie shell --seed 2024177 
(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)))