Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.6%

Time: 15.8s

Alternatives: 13

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: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \frac{-{c}^{2}}{{b}^{2}}\right) - c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -2.0
   (* (pow a 2.0) (/ (pow c 3.0) (pow b 4.0)))
   (-
    (fma
     -0.25
     (* (/ (* (pow a 4.0) (pow c 4.0)) a) (/ 20.0 (pow b 6.0)))
     (* a (/ (- (pow c 2.0)) (pow b 2.0))))
    c))
  b))

C

double code(double a, double b, double c) {
	return fma(-2.0, (pow(a, 2.0) * (pow(c, 3.0) / pow(b, 4.0))), (fma(-0.25, (((pow(a, 4.0) * pow(c, 4.0)) / a) * (20.0 / pow(b, 6.0))), (a * (-pow(c, 2.0) / pow(b, 2.0)))) - c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-2.0, Float64((a ^ 2.0) * Float64((c ^ 3.0) / (b ^ 4.0))), Float64(fma(-0.25, Float64(Float64(Float64((a ^ 4.0) * (c ^ 4.0)) / a) * Float64(20.0 / (b ^ 6.0))), Float64(a * Float64(Float64(-(c ^ 2.0)) / (b ^ 2.0)))) - c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[((-N[Power[c, 2.0], $MachinePrecision]) / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

   \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \frac{-{c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]
9. Add Preprocessing

Alternative 2: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{-5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} - \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -5.0 (/ (* (pow c 4.0) (pow a 3.0)) (pow b 6.0)))
   (-
    (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
    (+ c (/ (* a (pow c 2.0)) (pow b 2.0)))))
  b))

C

double code(double a, double b, double c) {
	return ((-5.0 * ((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 6.0))) + ((-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) - (c + ((a * pow(c, 2.0)) / pow(b, 2.0))))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-5.0d0) * (((c ** 4.0d0) * (a ** 3.0d0)) / (b ** 6.0d0))) + (((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 4.0d0))) - (c + ((a * (c ** 2.0d0)) / (b ** 2.0d0))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((-5.0 * ((Math.pow(c, 4.0) * Math.pow(a, 3.0)) / Math.pow(b, 6.0))) + ((-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 4.0))) - (c + ((a * Math.pow(c, 2.0)) / Math.pow(b, 2.0))))) / b;
}

Python

def code(a, b, c):
	return ((-5.0 * ((math.pow(c, 4.0) * math.pow(a, 3.0)) / math.pow(b, 6.0))) + ((-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 4.0))) - (c + ((a * math.pow(c, 2.0)) / math.pow(b, 2.0))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-5.0 * Float64(Float64((c ^ 4.0) * (a ^ 3.0)) / (b ^ 6.0))) + Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) - Float64(c + Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-5.0 * (((c ^ 4.0) * (a ^ 3.0)) / (b ^ 6.0))) + ((-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) - (c + ((a * (c ^ 2.0)) / (b ^ 2.0))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-5.0 * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in b around inf 98.2%

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

9. Final simplification 98.2%

   \[\leadsto \frac{-5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} - \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]
10. Add Preprocessing

Alternative 3: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\frac{{c}^{3} \cdot \left(-2 \cdot {a}^{2}\right)}{{b}^{4}} - c\right) - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right) + -5 \cdot \left({a}^{3} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (-
    (- (/ (* (pow c 3.0) (* -2.0 (pow a 2.0))) (pow b 4.0)) c)
    (* a (* (/ c b) (/ c b))))
   (* -5.0 (* (pow a 3.0) (/ (pow c 4.0) (pow b 6.0)))))
  b))

C

double code(double a, double b, double c) {
	return (((((pow(c, 3.0) * (-2.0 * pow(a, 2.0))) / pow(b, 4.0)) - c) - (a * ((c / b) * (c / b)))) + (-5.0 * (pow(a, 3.0) * (pow(c, 4.0) / pow(b, 6.0))))) / 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 ** 3.0d0) * ((-2.0d0) * (a ** 2.0d0))) / (b ** 4.0d0)) - c) - (a * ((c / b) * (c / b)))) + ((-5.0d0) * ((a ** 3.0d0) * ((c ** 4.0d0) / (b ** 6.0d0))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (((((Math.pow(c, 3.0) * (-2.0 * Math.pow(a, 2.0))) / Math.pow(b, 4.0)) - c) - (a * ((c / b) * (c / b)))) + (-5.0 * (Math.pow(a, 3.0) * (Math.pow(c, 4.0) / Math.pow(b, 6.0))))) / b;
}

Python

def code(a, b, c):
	return (((((math.pow(c, 3.0) * (-2.0 * math.pow(a, 2.0))) / math.pow(b, 4.0)) - c) - (a * ((c / b) * (c / b)))) + (-5.0 * (math.pow(a, 3.0) * (math.pow(c, 4.0) / math.pow(b, 6.0))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64((c ^ 3.0) * Float64(-2.0 * (a ^ 2.0))) / (b ^ 4.0)) - c) - Float64(a * Float64(Float64(c / b) * Float64(c / b)))) + Float64(-5.0 * Float64((a ^ 3.0) * Float64((c ^ 4.0) / (b ^ 6.0))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((((c ^ 3.0) * (-2.0 * (a ^ 2.0))) / (b ^ 4.0)) - c) - (a * ((c / b) * (c / b)))) + (-5.0 * ((a ^ 3.0) * ((c ^ 4.0) / (b ^ 6.0))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(-2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in b around inf 98.2%

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

   1. +-commutative 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. fma-define 98.2%

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

10. Applied egg-rr 98.2%

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

12. Simplified 98.2%

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

    1. unpow2 98.2%

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

    2. distribute-frac-neg2 98.2%

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

    3. distribute-frac-neg2 98.2%

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

14. Applied egg-rr 98.2%

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

15. Final simplification 98.2%

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

Alternative 4: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-1 + c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (+
    -1.0
    (*
     c
     (-
      (*
       c
       (+
        (* -5.0 (/ (* c (pow a 3.0)) (pow b 6.0)))
        (* -2.0 (/ (pow a 2.0) (pow b 4.0)))))
      (/ a (pow b 2.0))))))
  b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((c * ((-5.0 * ((c * pow(a, 3.0)) / pow(b, 6.0))) + (-2.0 * (pow(a, 2.0) / pow(b, 4.0))))) - (a / pow(b, 2.0)))))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c * ((-1.0d0) + (c * ((c * (((-5.0d0) * ((c * (a ** 3.0d0)) / (b ** 6.0d0))) + ((-2.0d0) * ((a ** 2.0d0) / (b ** 4.0d0))))) - (a / (b ** 2.0d0)))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((c * ((-5.0 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 6.0))) + (-2.0 * (Math.pow(a, 2.0) / Math.pow(b, 4.0))))) - (a / Math.pow(b, 2.0)))))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 + (c * ((c * ((-5.0 * ((c * math.pow(a, 3.0)) / math.pow(b, 6.0))) + (-2.0 * (math.pow(a, 2.0) / math.pow(b, 4.0))))) - (a / math.pow(b, 2.0)))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 + Float64(c * Float64(Float64(c * Float64(Float64(-5.0 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 6.0))) + Float64(-2.0 * Float64((a ^ 2.0) / (b ^ 4.0))))) - Float64(a / (b ^ 2.0)))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (-1.0 + (c * ((c * ((-5.0 * ((c * (a ^ 3.0)) / (b ^ 6.0))) + (-2.0 * ((a ^ 2.0) / (b ^ 4.0))))) - (a / (b ^ 2.0)))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-1.0 + N[(c * N[(N[(c * N[(N[(-5.0 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in b around inf 98.2%

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

   1. +-commutative 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. fma-define 98.2%

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

10. Applied egg-rr 98.2%

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

12. Simplified 98.2%

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

13. Taylor expanded in c around 0 98.2%

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

14. Final simplification 98.2%

    \[\leadsto \frac{c \cdot \left(-1 + c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right)\right)}{b} \]
15. Add Preprocessing

Alternative 5: 96.9% accurate, 0.2× speedup

Math

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

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 2.0)) (pow 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, 2.0)) / pow(b, 2.0)))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 4.0d0))) - (c + ((a * (c ** 2.0d0)) / (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, 2.0)) / Math.pow(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, 2.0)) / math.pow(b, 2.0)))) / b

Julia

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 97.5%

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

6. Final simplification 97.5%

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

Alternative 6: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-1 + c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (+
    -1.0
    (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 4.0))) (/ a (pow b 2.0))))))
  b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 4.0))) - (a / pow(b, 2.0)))))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c * ((-1.0d0) + (c * (((-2.0d0) * (((a ** 2.0d0) * c) / (b ** 4.0d0))) - (a / (b ** 2.0d0)))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((-2.0 * ((Math.pow(a, 2.0) * c) / Math.pow(b, 4.0))) - (a / Math.pow(b, 2.0)))))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 + (c * ((-2.0 * ((math.pow(a, 2.0) * c) / math.pow(b, 4.0))) - (a / math.pow(b, 2.0)))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 + Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0))) - Float64(a / (b ^ 2.0)))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (-1.0 + (c * ((-2.0 * (((a ^ 2.0) * c) / (b ^ 4.0))) - (a / (b ^ 2.0)))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-1.0 + N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in c around 0 97.5%

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

9. Final simplification 97.5%

   \[\leadsto \frac{c \cdot \left(-1 + c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right)\right)}{b} \]
10. Add Preprocessing

Alternative 7: 96.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in a around 0 97.5%

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

6. Taylor expanded in c around inf 97.5%

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

7. Final simplification 97.5%

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

Alternative 8: 96.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (- (* -2.0 (/ (* (pow a 2.0) c) (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 * ((pow(a, 2.0) * c) / 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) * (((a ** 2.0d0) * c) / (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 * ((Math.pow(a, 2.0) * c) / 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 * ((math.pow(a, 2.0) * c) / 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((a ^ 2.0) * c) / (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 * (((a ^ 2.0) * c) / (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[(N[Power[a, 2.0], $MachinePrecision] * c), $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 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in c around 0 97.1%

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

6. Final simplification 97.1%

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

Alternative 9: 95.3% 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(Float64(-(c ^ 2.0)) / (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 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

   3. mul-1-neg 95.9%

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

   4. associate-/l* 95.9%

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

7. Simplified 95.9%

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

8. Final simplification 95.9%

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

Alternative 10: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in c around 0 95.9%

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

   1. pow1 95.9%

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

   2. mul-1-neg 95.9%

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

   3. div-inv 95.9%

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

   4. pow-flip 95.9%

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

   5. metadata-eval 95.9%

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

10. Applied egg-rr 95.9%

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

    1. unpow1 95.9%

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

    2. distribute-lft-neg-in 95.9%

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

    3. *-commutative 95.9%

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

    4. distribute-rgt-neg-in 95.9%

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

12. Simplified 95.9%

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

13. Final simplification 95.9%

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

Alternative 11: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-1.0d0) / b) - (a * (c / (b ** 3.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in c around 0 95.5%

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

   1. sub-neg 95.5%

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

   2. mul-1-neg 95.5%

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

   3. distribute-neg-out 95.5%

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

   4. +-commutative 95.5%

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

   5. distribute-neg-out 95.5%

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

   6. unsub-neg 95.5%

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

   7. distribute-neg-frac 95.5%

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

   8. metadata-eval 95.5%

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

   9. associate-/l* 95.5%

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

10. Simplified 95.5%

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

Alternative 12: 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 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 90.7%

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

   1. associate-*r/ 90.7%

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

   2. mul-1-neg 90.7%

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

7. Simplified 90.7%

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

8. Final simplification 90.7%

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

Alternative 13: 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 17.7%

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

   1. *-commutative 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in b around inf 90.7%

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

   1. associate-*r/ 90.7%

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

   2. mul-1-neg 90.7%

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

7. Simplified 90.7%

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

   1. distribute-frac-neg 90.7%

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

   2. mul-1-neg 90.7%

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

   3. expm1-log1p-u 80.0%

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

   4. expm1-undefine 19.7%

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

   5. mul-1-neg 19.7%

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

   6. distribute-frac-neg 19.7%

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

9. Applied egg-rr 19.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-c}{b}\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 19.7%

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

    2. log1p-undefine 19.7%

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

    3. rem-exp-log 30.4%

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

    4. distribute-frac-neg 30.4%

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

    5. unsub-neg 30.4%

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

    6. metadata-eval 30.4%

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

11. Simplified 30.4%

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

12. Taylor expanded in c around 0 3.3%

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

13. Final simplification 3.3%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024156 
(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)))