Quadratic roots, wide range

Percentage Accurate: 17.6% → 97.8%

Time: 15.5s

Alternatives: 8

Speedup: 29.0×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $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 15.4%

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

   1. *-commutative 15.4%

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

   2. +-commutative 15.4%

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

   3. sqr-neg 15.4%

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

   4. unsub-neg 15.4%

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

   5. sqr-neg 15.4%

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

   6. fma-neg 15.4%

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

   7. distribute-lft-neg-in 15.4%

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

   8. *-commutative 15.4%

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

   9. *-commutative 15.4%

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

   10. distribute-rgt-neg-in 15.4%

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

   11. metadata-eval 15.4%

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

3. Simplified 15.4%

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

5. Taylor expanded in a around 0 98.6%

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

6. Taylor expanded in c around 0 98.6%

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

   1. associate-*r/ 98.6%

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

   2. neg-mul-1 98.6%

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

8. Applied egg-rr 98.6%

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

   1. associate-*r/ 98.6%

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

10. Applied egg-rr 98.6%

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

    1. neg-mul-1 98.6%

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

12. Simplified 98.6%

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

13. Final simplification 98.6%

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

Alternative 2: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(N[(-2.0 * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $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 15.4%

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

   1. *-commutative 15.4%

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

   2. +-commutative 15.4%

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

   3. sqr-neg 15.4%

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

   4. unsub-neg 15.4%

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

   5. sqr-neg 15.4%

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

   6. fma-neg 15.4%

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

   7. distribute-lft-neg-in 15.4%

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

   8. *-commutative 15.4%

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

   9. *-commutative 15.4%

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

   10. distribute-rgt-neg-in 15.4%

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

   11. metadata-eval 15.4%

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

3. Simplified 15.4%

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

5. Taylor expanded in b around inf 15.4%

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

   1. associate-*r/ 15.4%

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

   2. associate-*r* 15.4%

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

   3. *-commutative 15.4%

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

7. Simplified 15.4%

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

8. Taylor expanded in a around 0 98.1%

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

   1. neg-mul-1 98.1%

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

   2. distribute-frac-neg2 98.1%

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

   3. +-commutative 98.1%

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

   4. distribute-frac-neg2 98.1%

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

   5. unsub-neg 98.1%

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

   6. mul-1-neg 98.1%

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

   7. unsub-neg 98.1%

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

   8. associate-*r/ 98.1%

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

10. Simplified 98.1%

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

Alternative 3: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.4%

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

   1. *-commutative 15.4%

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

   2. +-commutative 15.4%

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

   3. sqr-neg 15.4%

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

   4. unsub-neg 15.4%

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

   5. sqr-neg 15.4%

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

   6. fma-neg 15.4%

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

   7. distribute-lft-neg-in 15.4%

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

   8. *-commutative 15.4%

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

   9. *-commutative 15.4%

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

   10. distribute-rgt-neg-in 15.4%

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

   11. metadata-eval 15.4%

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

3. Simplified 15.4%

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

5. Taylor expanded in b around inf 15.4%

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

   1. associate-*r/ 15.4%

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

   2. associate-*r* 15.4%

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

   3. *-commutative 15.4%

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

7. Simplified 15.4%

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

   1. clear-num 15.4%

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

   2. inv-pow 15.4%

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

   3. +-commutative 15.4%

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

   4. associate-/l* 15.4%

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

   5. fma-define 15.4%

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

9. Applied egg-rr 15.4%

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

    1. unpow-1 15.4%

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

    2. associate-/l* 15.4%

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

11. Simplified 15.4%

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

12. Taylor expanded in c around 0 97.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + c \cdot \left(-2 \cdot \left(c \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.5 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + \frac{a}{b}\right)}{c}}} \]
13. Step-by-step derivation

    1. neg-mul-1 97.9%

       \[\leadsto \frac{1}{\frac{\color{blue}{\left(-b\right)} + c \cdot \left(-2 \cdot \left(c \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.5 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + \frac{a}{b}\right)}{c}} \]

    2. +-commutative 97.9%

       \[\leadsto \frac{1}{\frac{\color{blue}{c \cdot \left(-2 \cdot \left(c \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.5 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + \frac{a}{b}\right) + \left(-b\right)}}{c}} \]

    3. unsub-neg 97.9%

       \[\leadsto \frac{1}{\frac{\color{blue}{c \cdot \left(-2 \cdot \left(c \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.5 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + \frac{a}{b}\right) - b}}{c}} \]

    4. fma-define 97.9%

       \[\leadsto \frac{1}{\frac{c \cdot \color{blue}{\mathsf{fma}\left(-2, c \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.5 \cdot \frac{{a}^{2}}{{b}^{3}}\right), \frac{a}{b}\right)} - b}{c}} \]

    5. distribute-rgt-out 97.9%

       \[\leadsto \frac{1}{\frac{c \cdot \mathsf{fma}\left(-2, c \cdot \color{blue}{\left(\frac{{a}^{2}}{{b}^{3}} \cdot \left(-1 + 0.5\right)\right)}, \frac{a}{b}\right) - b}{c}} \]

    6. metadata-eval 97.9%

       \[\leadsto \frac{1}{\frac{c \cdot \mathsf{fma}\left(-2, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot \color{blue}{-0.5}\right), \frac{a}{b}\right) - b}{c}} \]

14. Simplified 97.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \mathsf{fma}\left(-2, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot -0.5\right), \frac{a}{b}\right) - b}{c}}} \]

15. Taylor expanded in a around 0 97.9%

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

16. Final simplification 97.9%

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

Alternative 4: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.4%

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

   1. *-commutative 15.4%

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

   2. +-commutative 15.4%

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

   3. sqr-neg 15.4%

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

   4. unsub-neg 15.4%

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

   5. sqr-neg 15.4%

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

   6. fma-neg 15.4%

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

   7. distribute-lft-neg-in 15.4%

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

   8. *-commutative 15.4%

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

   9. *-commutative 15.4%

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

   10. distribute-rgt-neg-in 15.4%

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

   11. metadata-eval 15.4%

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

3. Simplified 15.4%

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

5. Taylor expanded in a around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. unsub-neg 96.7%

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

   3. mul-1-neg 96.7%

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

   4. distribute-neg-frac2 96.7%

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

   5. associate-/l* 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in b around inf 96.7%

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

   1. distribute-lft-out 96.7%

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

   2. associate-*r/ 96.7%

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

   3. mul-1-neg 96.7%

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

   4. associate-*r/ 96.7%

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

   5. distribute-neg-frac2 96.7%

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

10. Simplified 96.7%

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

    1. fma-undefine 96.7%

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

12. Applied egg-rr 96.7%

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

13. Final simplification 96.7%

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

Alternative 5: 95.2% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.4%

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

   1. *-commutative 15.4%

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

   2. +-commutative 15.4%

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

   3. sqr-neg 15.4%

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

   4. unsub-neg 15.4%

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

   5. sqr-neg 15.4%

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

   6. fma-neg 15.4%

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

   7. distribute-lft-neg-in 15.4%

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

   8. *-commutative 15.4%

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

   9. *-commutative 15.4%

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

   10. distribute-rgt-neg-in 15.4%

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

   11. metadata-eval 15.4%

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

3. Simplified 15.4%

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

5. Taylor expanded in b around inf 15.4%

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

   1. associate-*r/ 15.4%

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

   2. associate-*r* 15.4%

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

   3. *-commutative 15.4%

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

7. Simplified 15.4%

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

   1. clear-num 15.4%

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

   2. inv-pow 15.4%

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

   3. +-commutative 15.4%

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

   4. associate-/l* 15.4%

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

   5. fma-define 15.4%

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

9. Applied egg-rr 15.4%

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

    1. unpow-1 15.4%

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

    2. associate-/l* 15.4%

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

11. Simplified 15.4%

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

12. Taylor expanded in c around 0 96.6%

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

    1. neg-mul-1 96.6%

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

    2. +-commutative 96.6%

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

    3. unsub-neg 96.6%

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

    4. associate-/l* 96.6%

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

14. Simplified 96.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot \frac{c}{b} - b}{c}}} \]
15. Add Preprocessing

Alternative 6: 95.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.4%

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

   1. *-commutative 15.4%

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

   2. +-commutative 15.4%

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

   3. sqr-neg 15.4%

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

   4. unsub-neg 15.4%

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

   5. sqr-neg 15.4%

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

   6. fma-neg 15.4%

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

   7. distribute-lft-neg-in 15.4%

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

   8. *-commutative 15.4%

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

   9. *-commutative 15.4%

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

   10. distribute-rgt-neg-in 15.4%

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

   11. metadata-eval 15.4%

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

3. Simplified 15.4%

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

5. Taylor expanded in b around inf 15.4%

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

   1. associate-*r/ 15.4%

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

   2. associate-*r* 15.4%

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

   3. *-commutative 15.4%

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

7. Simplified 15.4%

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

   1. clear-num 15.4%

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

   2. inv-pow 15.4%

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

   3. +-commutative 15.4%

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

   4. associate-/l* 15.4%

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

   5. fma-define 15.4%

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

9. Applied egg-rr 15.4%

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

    1. unpow-1 15.4%

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

    2. associate-/l* 15.4%

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

11. Simplified 15.4%

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

12. Taylor expanded in a around 0 96.6%

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

    1. +-commutative 96.6%

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

    2. mul-1-neg 96.6%

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

    3. unsub-neg 96.6%

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

14. Simplified 96.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
15. Add Preprocessing

Alternative 7: 90.6% 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(Float64(-c) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.4%

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

   1. *-commutative 15.4%

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

   2. +-commutative 15.4%

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

   3. sqr-neg 15.4%

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

   4. unsub-neg 15.4%

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

   5. sqr-neg 15.4%

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

   6. fma-neg 15.4%

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

   7. distribute-lft-neg-in 15.4%

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

   8. *-commutative 15.4%

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

   9. *-commutative 15.4%

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

   10. distribute-rgt-neg-in 15.4%

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

   11. metadata-eval 15.4%

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

3. Simplified 15.4%

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

5. Taylor expanded in b around inf 92.6%

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

   1. associate-*r/ 92.6%

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

   2. mul-1-neg 92.6%

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

7. Simplified 92.6%

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

Alternative 8: 1.7% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.4%

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

   1. *-commutative 15.4%

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

   2. +-commutative 15.4%

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

   3. sqr-neg 15.4%

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

   4. unsub-neg 15.4%

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

   5. sqr-neg 15.4%

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

   6. fma-neg 15.4%

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

   7. distribute-lft-neg-in 15.4%

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

   8. *-commutative 15.4%

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

   9. *-commutative 15.4%

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

   10. distribute-rgt-neg-in 15.4%

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

   11. metadata-eval 15.4%

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

3. Simplified 15.4%

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

5. Taylor expanded in b around inf 92.6%

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

   1. associate-*r/ 92.6%

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

   2. mul-1-neg 92.6%

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

7. Simplified 92.6%

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

   1. distribute-frac-neg 92.6%

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

   2. distribute-frac-neg2 92.6%

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

   3. clear-num 92.3%

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

   4. inv-pow 92.3%

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

9. Applied egg-rr 92.3%

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

    1. unpow-1 92.3%

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

    2. distribute-frac-neg 92.3%

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

    3. distribute-frac-neg2 92.3%

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

11. Simplified 92.3%

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

    1. clear-num 92.6%

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

    2. add-sqr-sqrt 0.0%

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

    3. sqrt-unprod 1.7%

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

    4. sqr-neg 1.7%

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

    5. sqrt-unprod 1.7%

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

    6. add-sqr-sqrt 1.7%

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

    7. *-un-lft-identity 1.7%

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

13. Applied egg-rr 1.7%

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

    1. *-lft-identity 1.7%

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

15. Simplified 1.7%

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

Reproduce

herbie shell --seed 2024145 
(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)))