Quadratic roots, medium range

Percentage Accurate: 32.3% → 95.4%

Time: 16.5s

Alternatives: 8

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 32.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \frac{c}{b} \cdot \frac{c}{b}, c\right), -0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \frac{20}{a}}{{b}^{6}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -2.0
   (* (pow a 2.0) (/ (pow c 3.0) (pow b 4.0)))
   (fma
    -1.0
    (fma a (* (/ c b) (/ c b)) c)
    (* -0.25 (/ (* (pow (* a c) 4.0) (/ 20.0 a)) (pow b 6.0)))))
  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(-1.0, fma(a, ((c / b) * (c / b)), c), (-0.25 * ((pow((a * c), 4.0) * (20.0 / a)) / pow(b, 6.0))))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-2.0, Float64((a ^ 2.0) * Float64((c ^ 3.0) / (b ^ 4.0))), fma(-1.0, fma(a, Float64(Float64(c / b) * Float64(c / b)), c), Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) * Float64(20.0 / a)) / (b ^ 6.0))))) / 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[(-1.0 * N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] + N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.2%

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

   1. *-commutative 31.2%

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

3. Simplified 31.2%

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

5. Taylor expanded in b around inf 31.2%

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

6. Taylor expanded in b around inf 95.6%

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

8. Simplified 95.6%

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

   1. associate-*l/ 95.6%

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

   2. pow-prod-down 95.6%

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

10. Applied egg-rr 95.6%

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

    1. unpow2 95.6%

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

12. Applied egg-rr 95.6%

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

13. Final simplification 95.6%

    \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \frac{c}{b} \cdot \frac{c}{b}, c\right), -0.25 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot \frac{20}{a}}{{b}^{6}}\right)\right)}{b} \]
14. Add Preprocessing

Alternative 2: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \frac{20 \cdot {c}^{4}}{{b}^{6}}}{b}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

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 6.0))) b))))
    (/ (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, 6.0))) / b)))) - (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 ** 6.0d0))) / b)))) - ((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, 6.0))) / b)))) - (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, 6.0))) / b)))) - (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(Float64(a * Float64(Float64(20.0 * (c ^ 4.0)) / (b ^ 6.0))) / b)))) - 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 ^ 6.0))) / b)))) - ((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[(N[(a * N[(N[(20.0 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $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 31.2%

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

   1. *-commutative 31.2%

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

3. Simplified 31.2%

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

5. Taylor expanded in a around 0 95.5%

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

6. Taylor expanded in c around 0 95.5%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \color{blue}{\left(20 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}}{b}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 95.5%

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

   2. associate-*l/ 95.5%

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

8. Simplified 95.5%

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

9. Final simplification 95.5%

   \[\leadsto a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \frac{20 \cdot {c}^{4}}{{b}^{6}}}{b}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
10. Add Preprocessing

Alternative 3: 93.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

   1. *-commutative 31.2%

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

3. Simplified 31.2%

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

5. Taylor expanded in b around inf 94.0%

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

   1. associate-+r+ 94.0%

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

   2. mul-1-neg 94.0%

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

   3. unsub-neg 94.0%

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

   4. mul-1-neg 94.0%

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

   5. unsub-neg 94.0%

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

   6. associate-/l* 94.0%

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

   7. associate-*r* 94.0%

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

7. Simplified 94.0%

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

   1. associate-/l* 94.0%

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

9. Applied egg-rr 94.0%

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

    1. unpow2 94.0%

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

    2. unpow2 94.0%

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

    3. times-frac 94.0%

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

    4. sqr-neg 94.0%

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

    5. unpow2 94.0%

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

    6. distribute-neg-frac2 94.0%

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

11. Simplified 94.0%

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

    1. unpow2 95.6%

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

13. Applied egg-rr 94.0%

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

14. Final simplification 94.0%

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

Alternative 4: 93.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (c * ((a / pow(b, 3.0)) - (-2.0 * ((pow(a, 2.0) * c) / 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 * (((-1.0d0) / b) - (c * ((a / (b ** 3.0d0)) - ((-2.0d0) * (((a ** 2.0d0) * c) / (b ** 5.0d0))))))
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)) - (-2.0 * ((Math.pow(a, 2.0) * c) / Math.pow(b, 5.0))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

   1. *-commutative 31.2%

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

3. Simplified 31.2%

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

5. Taylor expanded in c around 0 93.7%

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

6. Final simplification 93.7%

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

Alternative 5: 90.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

   1. *-commutative 31.2%

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

3. Simplified 31.2%

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

5. Taylor expanded in a around 0 90.9%

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

   1. mul-1-neg 90.9%

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

   2. unsub-neg 90.9%

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

   3. mul-1-neg 90.9%

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

   4. distribute-neg-frac2 90.9%

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

   5. associate-/l* 90.9%

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

7. Simplified 90.9%

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

   1. expm1-log1p-u 80.2%

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

   2. distribute-frac-neg2 80.2%

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

   3. *-commutative 80.2%

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

   4. div-inv 80.2%

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

   5. pow-flip 80.2%

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

   6. metadata-eval 80.2%

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

9. Applied egg-rr 80.2%

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

    1. expm1-undefine 37.7%

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

    2. sub-neg 37.7%

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

    3. log1p-undefine 37.7%

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

    4. rem-exp-log 48.4%

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

    5. sub-neg 48.4%

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

    6. distribute-neg-out 48.4%

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

    7. unsub-neg 48.4%

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

    8. associate-*l* 48.4%

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

    9. metadata-eval 48.4%

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

11. Simplified 48.4%

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

12. Taylor expanded in b around inf 91.0%

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

    1. sub-neg 91.0%

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

    2. mul-1-neg 91.0%

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

    3. +-commutative 91.0%

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

    4. distribute-lft-out 91.0%

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

    5. mul-1-neg 91.0%

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

    6. +-commutative 91.0%

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

    7. remove-double-neg 91.0%

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

    8. mul-1-neg 91.0%

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

    9. sub-neg 91.0%

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

    10. distribute-neg-frac 91.0%

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

    11. distribute-neg-frac2 91.0%

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

14. Simplified 91.0%

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

Alternative 6: 90.4% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 31.2%

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

   1. *-commutative 31.2%

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

3. Simplified 31.2%

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

5. Taylor expanded in c around 0 90.7%

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

   1. associate-*r/ 90.7%

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

   2. neg-mul-1 90.7%

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

   3. distribute-rgt-neg-in 90.7%

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

7. Simplified 90.7%

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

8. Final simplification 90.7%

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

Alternative 7: 80.7% 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 31.2%

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

   1. *-commutative 31.2%

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

3. Simplified 31.2%

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

5. Taylor expanded in b around inf 81.3%

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

   1. associate-*r/ 81.3%

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

   2. mul-1-neg 81.3%

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

7. Simplified 81.3%

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

8. Final simplification 81.3%

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

Alternative 8: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

   1. *-commutative 31.2%

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

3. Simplified 31.2%

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

5. Taylor expanded in b around inf 31.2%

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

   1. log1p-expm1-u 22.7%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-b\right) + \sqrt{{b}^{2} \cdot \left(1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\right)\right)}}{a \cdot 2} \]

   2. neg-mul-1 22.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-1 \cdot b} + \sqrt{{b}^{2} \cdot \left(1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\right)\right)}{a \cdot 2} \]

   3. fma-define 22.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} \cdot \left(1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\right)}\right)\right)}{a \cdot 2} \]

   4. +-commutative 22.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} \cdot \color{blue}{\left(-4 \cdot \frac{a \cdot c}{{b}^{2}} + 1\right)}}\right)\right)\right)}{a \cdot 2} \]

   5. fma-define 22.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} \cdot \color{blue}{\mathsf{fma}\left(-4, \frac{a \cdot c}{{b}^{2}}, 1\right)}}\right)\right)\right)}{a \cdot 2} \]

   6. div-inv 22.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} \cdot \mathsf{fma}\left(-4, \color{blue}{\left(a \cdot c\right) \cdot \frac{1}{{b}^{2}}}, 1\right)}\right)\right)\right)}{a \cdot 2} \]

   7. *-commutative 22.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} \cdot \mathsf{fma}\left(-4, \color{blue}{\left(c \cdot a\right)} \cdot \frac{1}{{b}^{2}}, 1\right)}\right)\right)\right)}{a \cdot 2} \]

   8. pow-flip 22.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} \cdot \mathsf{fma}\left(-4, \left(c \cdot a\right) \cdot \color{blue}{{b}^{\left(-2\right)}}, 1\right)}\right)\right)\right)}{a \cdot 2} \]

   9. metadata-eval 22.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} \cdot \mathsf{fma}\left(-4, \left(c \cdot a\right) \cdot {b}^{\color{blue}{-2}}, 1\right)}\right)\right)\right)}{a \cdot 2} \]

7. Applied egg-rr 22.7%

   \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} \cdot \mathsf{fma}\left(-4, \left(c \cdot a\right) \cdot {b}^{-2}, 1\right)}\right)\right)\right)}}{a \cdot 2} \]

8. Taylor expanded in c around 0 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

10. Simplified 3.2%

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

Reproduce

herbie shell --seed 2024108 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))