Quadratic roots, wide range

Percentage Accurate: 17.7% → 97.4%

Time: 12.4s

Alternatives: 6

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - c \cdot \left(a \cdot \left({b}^{-3} - a \cdot \left(-5 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{7}} + -2 \cdot \frac{c}{{b}^{5}}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (-
   (/ -1.0 b)
   (*
    c
    (*
     a
     (-
      (pow b -3.0)
      (*
       a
       (+
        (* -5.0 (/ (* a (* c c)) (pow b 7.0)))
        (* -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) - (a * ((-5.0 * ((a * (c * c)) / pow(b, 7.0))) + (-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)) - (a * (((-5.0d0) * ((a * (c * c)) / (b ** 7.0d0))) + ((-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) - (a * ((-5.0 * ((a * (c * c)) / Math.pow(b, 7.0))) + (-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) - (a * ((-5.0 * ((a * (c * c)) / math.pow(b, 7.0))) + (-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(a * Float64((b ^ -3.0) - Float64(a * Float64(Float64(-5.0 * Float64(Float64(a * Float64(c * c)) / (b ^ 7.0))) + Float64(-2.0 * Float64(c / (b ^ 5.0))))))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - (c * (a * ((b ^ -3.0) - (a * ((-5.0 * ((a * (c * c)) / (b ^ 7.0))) + (-2.0 * (c / (b ^ 5.0)))))))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(c * N[(a * N[(N[Power[b, -3.0], $MachinePrecision] - N[(a * N[(N[(-5.0 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

   1. *-commutative 19.5%

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

3. Simplified 19.5%

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

5. Taylor expanded in c around 0 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in a around 0 96.4%

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

   1. unpow2 96.4%

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

10. Applied egg-rr 96.4%

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

    1. *-un-lft-identity 96.4%

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

    2. pow-flip 96.4%

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

    3. metadata-eval 96.4%

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

12. Applied egg-rr 96.4%

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

    1. *-lft-identity 96.4%

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

14. Simplified 96.4%

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

15. Final simplification 96.4%

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

Alternative 2: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot a}{b}\\ c \cdot \left(\frac{-1}{b} - \frac{c \cdot a - -2 \cdot \left(t\_0 \cdot t\_0\right)}{{b}^{3}}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c a) b)))
   (* c (- (/ -1.0 b) (/ (- (* c a) (* -2.0 (* t_0 t_0))) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) / b;
	return c * ((-1.0 / b) - (((c * a) - (-2.0 * (t_0 * t_0))) / 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
    real(8) :: t_0
    t_0 = (c * a) / b
    code = c * (((-1.0d0) / b) - (((c * a) - ((-2.0d0) * (t_0 * t_0))) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * a) / b;
	return c * ((-1.0 / b) - (((c * a) - (-2.0 * (t_0 * t_0))) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	t_0 = (c * a) / b
	return c * ((-1.0 / b) - (((c * a) - (-2.0 * (t_0 * t_0))) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) / b)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(Float64(c * a) - Float64(-2.0 * Float64(t_0 * t_0))) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (c * a) / b;
	tmp = c * ((-1.0 / b) - (((c * a) - (-2.0 * (t_0 * t_0))) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]}, N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(N[(c * a), $MachinePrecision] - N[(-2.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.5%

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

   1. *-commutative 19.5%

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

3. Simplified 19.5%

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

5. Taylor expanded in c around 0 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in b around inf 95.8%

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

   1. mul-1-neg 95.8%

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

   2. unsub-neg 95.8%

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

   3. associate-/l* 95.8%

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

   4. unpow2 95.8%

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

   5. unpow2 95.8%

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

   6. unpow2 95.8%

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

   7. times-frac 95.8%

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

   8. swap-sqr 95.8%

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

   9. unpow2 95.8%

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

10. Simplified 95.8%

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

    1. unpow2 95.8%

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

    2. associate-*r/ 95.8%

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

    3. associate-*r/ 95.8%

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

12. Applied egg-rr 95.8%

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

13. Final simplification 95.8%

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

Alternative 3: 95.4% accurate, 1.0× speedup

Math

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

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(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / 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 19.5%

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

   1. *-commutative 19.5%

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

3. Simplified 19.5%

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

5. Taylor expanded in b around inf 94.7%

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

   1. mul-1-neg 94.7%

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

   2. unsub-neg 94.7%

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

   3. mul-1-neg 94.7%

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

   4. associate-/l* 94.7%

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

7. Simplified 94.7%

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

8. Taylor expanded in a around 0 94.7%

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

   1. associate-/l* 94.7%

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

   2. unpow2 94.7%

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

   3. unpow2 94.7%

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

   4. times-frac 94.7%

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

   5. sqr-neg 94.7%

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

   6. distribute-frac-neg 94.7%

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

   7. distribute-frac-neg 94.7%

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

   8. unpow2 94.7%

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

   9. distribute-frac-neg 94.7%

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

   10. distribute-neg-frac2 94.7%

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

10. Simplified 94.7%

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

Alternative 4: 95.2% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

   1. *-commutative 19.5%

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

3. Simplified 19.5%

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

5. Taylor expanded in b around inf 94.7%

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

   1. mul-1-neg 94.7%

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

   2. unsub-neg 94.7%

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

   3. mul-1-neg 94.7%

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

   4. associate-/l* 94.7%

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

7. Simplified 94.7%

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

8. Taylor expanded in a around inf 94.5%

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

   1. mul-1-neg 94.5%

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

   2. distribute-neg-frac2 94.5%

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

   3. unpow2 94.5%

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

   4. unpow2 94.5%

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

   5. times-frac 94.5%

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

   6. sqr-neg 94.5%

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

   7. distribute-frac-neg 94.5%

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

   8. distribute-frac-neg 94.5%

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

   9. unpow2 94.5%

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

   10. distribute-frac-neg 94.5%

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

   11. distribute-neg-frac2 94.5%

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

10. Simplified 94.5%

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

    1. unpow2 94.5%

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

12. Applied egg-rr 94.5%

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

13. Final simplification 94.5%

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

Alternative 5: 94.9% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

   1. *-commutative 19.5%

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

3. Simplified 19.5%

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

5. Taylor expanded in b around inf 94.7%

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

   1. mul-1-neg 94.7%

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

   2. unsub-neg 94.7%

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

   3. mul-1-neg 94.7%

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

   4. associate-/l* 94.7%

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

7. Simplified 94.7%

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

8. Taylor expanded in a around inf 94.5%

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

   1. mul-1-neg 94.5%

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

   2. distribute-neg-frac2 94.5%

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

   3. unpow2 94.5%

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

   4. unpow2 94.5%

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

   5. times-frac 94.5%

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

   6. sqr-neg 94.5%

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

   7. distribute-frac-neg 94.5%

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

   8. distribute-frac-neg 94.5%

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

   9. unpow2 94.5%

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

   10. distribute-frac-neg 94.5%

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

   11. distribute-neg-frac2 94.5%

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

10. Simplified 94.5%

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

    1. associate-/l* 94.3%

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

12. Applied egg-rr 94.3%

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

    1. unpow2 94.5%

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

14. Applied egg-rr 94.3%

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

15. Final simplification 94.3%

    \[\leadsto a \cdot \frac{\frac{c}{-a} - \frac{c}{b} \cdot \frac{c}{b}}{b} \]
16. Add Preprocessing

Alternative 6: 90.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

   1. *-commutative 19.5%

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

3. Simplified 19.5%

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

5. Taylor expanded in b around inf 89.3%

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

   1. associate-*r/ 89.3%

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

   2. mul-1-neg 89.3%

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

7. Simplified 89.3%

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

8. Final simplification 89.3%

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

Reproduce

herbie shell --seed 2024182 
(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)))