Quadratic roots, wide range

Percentage Accurate: 17.7% → 97.7%

Time: 14.1s

Alternatives: 9

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: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-5.0 * N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $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 20.7%

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

   1. *-commutative 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in a around 0 96.8%

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

6. Taylor expanded in c around 0 96.8%

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

   1. associate-*r/ 96.8%

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

   2. *-commutative 96.8%

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

8. Simplified 96.8%

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

   1. associate-*r/ 96.8%

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

10. Applied egg-rr 96.8%

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

    1. mul-1-neg 96.8%

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

12. Simplified 96.8%

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

    1. mul-1-neg 96.8%

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

14. Applied egg-rr 96.8%

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

15. Final simplification 96.8%

    \[\leadsto a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-5 \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
16. 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 20.7%

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

   1. *-commutative 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in c around 0 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in a around 0 96.4%

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

9. Final simplification 96.4%

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

Alternative 3: 96.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((a * ((-2.0 * (a * (pow(c, 3.0) / pow(b, 4.0)))) - pow((c / -b), 2.0))) / b) - (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 ** 4.0d0)))) - ((c / -b) ** 2.0d0))) / b) - (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, 4.0)))) - Math.pow((c / -b), 2.0))) / b) - (c / b);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.7%

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

   1. *-commutative 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in b around inf 95.6%

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

6. Taylor expanded in a around 0 95.6%

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

   1. neg-mul-1 95.6%

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

   2. +-commutative 95.6%

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

   3. unsub-neg 95.6%

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

   4. mul-1-neg 95.6%

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

   5. unsub-neg 95.6%

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

   6. unpow2 95.6%

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

   7. unpow2 95.6%

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

   8. times-frac 95.6%

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

   9. sqr-neg 95.6%

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

   10. distribute-frac-neg 95.6%

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

   11. distribute-frac-neg 95.6%

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

   12. unpow2 95.6%

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

8. Simplified 95.6%

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

   1. div-sub 95.6%

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

   2. *-commutative 95.6%

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

   3. associate-/l* 95.6%

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

10. Applied egg-rr 95.6%

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

11. Final simplification 95.6%

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

Alternative 4: 96.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 4.0))) - ((c / b) * (c / b)))) - 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 ** 4.0d0))) - ((c / b) * (c / b)))) - 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, 4.0))) - ((c / b) * (c / b)))) - c) / b;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.7%

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

   1. *-commutative 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in b around inf 95.6%

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

6. Taylor expanded in a around 0 95.6%

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

   1. neg-mul-1 95.6%

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

   2. +-commutative 95.6%

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

   3. unsub-neg 95.6%

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

   4. mul-1-neg 95.6%

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

   5. unsub-neg 95.6%

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

   6. unpow2 95.6%

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

   7. unpow2 95.6%

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

   8. times-frac 95.6%

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

   9. sqr-neg 95.6%

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

   10. distribute-frac-neg 95.6%

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

   11. distribute-frac-neg 95.6%

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

   12. unpow2 95.6%

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

8. Simplified 95.6%

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

   1. unpow2 95.6%

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

10. Applied egg-rr 95.6%

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

11. Final simplification 95.6%

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

Alternative 5: 96.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (* a (+ (* -2.0 (/ (* c a) (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 * ((-2.0 * ((c * a) / 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 * (((-2.0d0) * ((c * a) / (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 * ((-2.0 * ((c * a) / 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 * ((-2.0 * ((c * a) / 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(-2.0 * Float64(Float64(c * a) / (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 * ((-2.0 * ((c * a) / (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[(-2.0 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 5.0], $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 20.7%

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

   1. *-commutative 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in c around 0 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in a around 0 95.3%

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

   1. *-commutative 95.3%

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

10. Simplified 95.3%

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

11. Final simplification 95.3%

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

Alternative 6: 95.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \frac{{c}^{2}}{-{b}^{3}} - \frac{c}{b} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return (a * (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 * ((c ** 2.0d0) / -(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) / -Math.pow(b, 3.0))) - (c / b);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.7%

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

   1. *-commutative 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in a around 0 93.7%

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

   1. mul-1-neg 93.7%

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

   2. unsub-neg 93.7%

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

   3. mul-1-neg 93.7%

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

   4. distribute-neg-frac2 93.7%

      \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

   5. associate-/l* 93.7%

      \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

7. Simplified 93.7%

   \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

8. Final simplification 93.7%

   \[\leadsto a \cdot \frac{{c}^{2}}{-{b}^{3}} - \frac{c}{b} \]
9. Add Preprocessing

Alternative 7: 95.3% 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 / 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 20.7%

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

   1. *-commutative 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in b around inf 95.6%

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

6. Taylor expanded in a around 0 95.6%

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

   1. neg-mul-1 95.6%

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

   2. +-commutative 95.6%

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

   3. unsub-neg 95.6%

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

   4. mul-1-neg 95.6%

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

   5. unsub-neg 95.6%

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

   6. unpow2 95.6%

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

   7. unpow2 95.6%

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

   8. times-frac 95.6%

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

   9. sqr-neg 95.6%

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

   10. distribute-frac-neg 95.6%

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

   11. distribute-frac-neg 95.6%

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

   12. unpow2 95.6%

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

8. Simplified 95.6%

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

9. Taylor expanded in a around 0 93.7%

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

    1. mul-1-neg 93.7%

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

    2. associate-/l* 93.7%

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

    3. unpow2 93.7%

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

    4. unpow2 93.7%

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

    5. times-frac 93.7%

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

    6. sqr-neg 93.7%

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

    7. distribute-frac-neg 93.7%

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

    8. distribute-frac-neg 93.7%

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

    9. unpow2 93.7%

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

    10. distribute-lft-neg-in 93.7%

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

    11. unpow2 93.7%

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

    12. distribute-frac-neg 93.7%

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

    13. distribute-frac-neg 93.7%

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

    14. sqr-neg 93.7%

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

    15. unpow2 93.7%

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

11. Simplified 93.7%

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

12. Final simplification 93.7%

    \[\leadsto \frac{c + a \cdot {\left(\frac{c}{b}\right)}^{2}}{-b} \]
13. 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 20.7%

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

   1. *-commutative 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in b around inf 88.3%

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

   1. associate-*r/ 88.3%

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

   2. mul-1-neg 88.3%

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

7. Simplified 88.3%

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

8. Final simplification 88.3%

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

Alternative 9: 1.7% accurate, 38.7× 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 / 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 20.7%

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

   1. *-commutative 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in b around -inf 8.9%

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

   1. mul-1-neg 8.9%

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

   2. *-commutative 8.9%

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

   3. distribute-rgt-neg-in 8.9%

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

   4. +-commutative 8.9%

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

   5. mul-1-neg 8.9%

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

   6. unsub-neg 8.9%

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

7. Simplified 8.9%

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

8. Taylor expanded in a around inf 1.7%

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

Reproduce

herbie shell --seed 2024131 
(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)))