Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.7%

Time: 14.0s

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: 17.8% 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: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(a \cdot \frac{20 \cdot {c}^{4}}{{b}^{7}}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (+
      (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
      (* -0.25 (* a (/ (* 20.0 (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))) + (-0.25 * (a * ((20.0 * 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))) + ((-0.25d0) * (a * ((20.0d0 * (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))) + (-0.25 * (a * ((20.0 * 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))) + (-0.25 * (a * ((20.0 * 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(-0.25 * Float64(a * Float64(Float64(20.0 * (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))) + (-0.25 * (a * ((20.0 * (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[(-0.25 * N[(a * N[(N[(20.0 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $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 14.9%

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

   1. *-commutative 14.9%

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

3. Simplified 14.9%

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

5. Taylor expanded in a around 0 99.0%

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

6. Taylor expanded in c around 0 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*l/ 99.0%

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

   3. associate-*r* 99.0%

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

   4. metadata-eval 99.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left({c}^{4} \cdot \color{blue}{\left(4 + 16\right)}\right)}{{b}^{7}}\right)\right) \]

   5. distribute-rgt-out 99.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \color{blue}{\left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}}{{b}^{7}}\right)\right) \]

   6. associate-/l* 99.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\left(a \cdot \frac{4 \cdot {c}^{4} + 16 \cdot {c}^{4}}{{b}^{7}}\right)}\right)\right) \]

   7. distribute-rgt-out 99.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(a \cdot \frac{\color{blue}{{c}^{4} \cdot \left(4 + 16\right)}}{{b}^{7}}\right)\right)\right) \]

   8. metadata-eval 99.0%

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

   9. *-commutative 99.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

   \[\leadsto a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(a \cdot \frac{20 \cdot {c}^{4}}{{b}^{7}}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
10. Add Preprocessing

Alternative 2: 96.9% accurate, 0.3× speedup

Math

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

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(Float64(-2.0 * (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[(N[(-2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 14.9%

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

   1. *-commutative 14.9%

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

3. Simplified 14.9%

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

5. Taylor expanded in b around inf 98.5%

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

   1. associate-+r+ 98.5%

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

   2. mul-1-neg 98.5%

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

   3. unsub-neg 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. associate-/l* 98.5%

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

   7. associate-*r* 98.5%

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

   8. associate-/l* 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in a around 0 98.5%

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

   1. associate-/l* 98.5%

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

   2. unpow2 98.5%

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

   3. unpow2 98.5%

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

   4. times-frac 98.5%

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

   5. sqr-neg 98.5%

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

   6. distribute-frac-neg 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. unpow2 98.5%

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

   9. distribute-frac-neg 98.5%

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

   10. distribute-neg-frac2 98.5%

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

10. Simplified 98.5%

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

11. Final simplification 98.5%

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

Alternative 3: 96.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

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 / -b) - (a * (((c ** 2.0d0) / (b ** 3.0d0)) + (2.0d0 * ((a * (c ** 3.0d0)) / (b ** 5.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.9%

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

   1. *-commutative 14.9%

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

3. Simplified 14.9%

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

5. Taylor expanded in c around 0 98.2%

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

6. Taylor expanded in c around -inf 97.9%

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

7. Taylor expanded in a around 0 98.5%

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

8. Final simplification 98.5%

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

Alternative 4: 96.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (fma(2.0, pow((c * (a / b)), 2.0), (c * a)) / pow(b, 3.0)));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 14.9%

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

   1. *-commutative 14.9%

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

3. Simplified 14.9%

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

5. Taylor expanded in c around 0 98.2%

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

6. Taylor expanded in b around -inf 98.2%

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

   1. mul-1-neg 98.2%

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

   2. distribute-neg-frac2 98.2%

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

   3. fma-define 98.2%

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

   4. associate-/l* 98.2%

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

   5. unpow2 98.2%

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

   6. unpow2 98.2%

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

   7. unpow2 98.2%

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

   8. times-frac 98.2%

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

   9. swap-sqr 98.2%

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

   10. unpow2 98.2%

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

   11. associate-*r/ 98.2%

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

   12. *-commutative 98.2%

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

   13. associate-/l* 98.2%

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

   14. *-commutative 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 5: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.9%

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

   1. *-commutative 14.9%

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

3. Simplified 14.9%

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

5. Taylor expanded in a around 0 97.1%

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

   1. mul-1-neg 97.1%

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

   2. unsub-neg 97.1%

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

   3. associate-*r/ 97.1%

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

   4. mul-1-neg 97.1%

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

   5. associate-/l* 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in a around inf 96.7%

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

   1. associate-*r/ 96.7%

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

   2. neg-mul-1 96.7%

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

10. Simplified 96.7%

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

11. Taylor expanded in b around inf 96.7%

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

    1. distribute-lft-out 96.7%

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

    2. associate-*r/ 96.7%

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

    3. mul-1-neg 96.7%

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

    4. distribute-neg-frac2 96.7%

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

    5. unpow2 96.7%

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

    6. unpow2 96.7%

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

    7. times-frac 96.7%

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

    8. unpow2 96.7%

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

13. Simplified 96.7%

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

14. Final simplification 96.7%

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

Alternative 6: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.9%

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

   1. *-commutative 14.9%

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

3. Simplified 14.9%

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

5. Taylor expanded in a around 0 97.1%

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

   1. mul-1-neg 97.1%

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

   2. unsub-neg 97.1%

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

   3. associate-*r/ 97.1%

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

   4. mul-1-neg 97.1%

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

   5. associate-/l* 97.1%

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

7. Simplified 97.1%

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

   1. unpow2 97.1%

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

9. Applied egg-rr 97.1%

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

10. Final simplification 97.1%

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

Alternative 7: 94.9% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 14.9%

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

   1. *-commutative 14.9%

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

3. Simplified 14.9%

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

5. Taylor expanded in c around 0 96.8%

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

   1. associate-*r/ 96.8%

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

   2. neg-mul-1 96.8%

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

   3. distribute-rgt-neg-in 96.8%

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

7. Simplified 96.8%

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

8. Final simplification 96.8%

   \[\leadsto c \cdot \left(\frac{-1}{b} - \frac{c \cdot a}{{b}^{3}}\right) \]
9. 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 14.9%

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

   1. *-commutative 14.9%

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

3. Simplified 14.9%

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

5. Taylor expanded in b around inf 92.5%

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

   1. associate-*r/ 92.5%

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

   2. mul-1-neg 92.5%

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

7. Simplified 92.5%

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

8. Final simplification 92.5%

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

Alternative 9: 3.3% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 14.9%

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

   1. *-commutative 14.9%

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

3. Simplified 14.9%

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

5. Taylor expanded in a around 0 97.1%

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

   1. mul-1-neg 97.1%

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

   2. unsub-neg 97.1%

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

   3. associate-*r/ 97.1%

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

   4. mul-1-neg 97.1%

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

   5. associate-/l* 97.1%

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

7. Simplified 97.1%

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

   1. expm1-log1p-u 84.6%

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

   2. *-commutative 84.6%

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

   3. div-inv 84.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-c}{b} - \color{blue}{\left({c}^{2} \cdot \frac{1}{{b}^{3}}\right)} \cdot a\right)\right) \]

   4. pow-flip 84.6%

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

   5. metadata-eval 84.6%

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

9. Applied egg-rr 84.6%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-c}{b} - \left({c}^{2} \cdot {b}^{-3}\right) \cdot a\right)\right)} \]
10. Step-by-step derivation

    1. expm1-undefine 21.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-c}{b} - \left({c}^{2} \cdot {b}^{-3}\right) \cdot a\right)} - 1} \]

    2. sub-neg 21.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-c}{b} - \left({c}^{2} \cdot {b}^{-3}\right) \cdot a\right)} + \left(-1\right)} \]

    3. log1p-undefine 21.3%

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

    4. rem-exp-log 33.9%

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

    5. sub-neg 33.9%

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

    6. distribute-frac-neg 33.9%

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

    7. distribute-neg-out 33.9%

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

    8. unsub-neg 33.9%

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

    9. *-commutative 33.9%

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

    10. metadata-eval 33.9%

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

11. Simplified 33.9%

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

12. Taylor expanded in c around 0 31.8%

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

13. Taylor expanded in c around 0 3.3%

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

14. Final simplification 3.3%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(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)))