Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.8%

Time: 21.9s

Alternatives: 5

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -2.0
  (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0))
  (-
   (* -0.25 (/ (* 20.0 (pow (* c a) 4.0)) (* a (pow b 7.0))))
   (fma a (/ (pow c 2.0) (pow b 3.0)) (/ c b)))))

C

double code(double a, double b, double c) {
	return fma(-2.0, ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0)), ((-0.25 * ((20.0 * pow((c * a), 4.0)) / (a * pow(b, 7.0)))) - fma(a, (pow(c, 2.0) / pow(b, 3.0)), (c / b))));
}

Julia

function code(a, b, c)
	return fma(-2.0, Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0)), Float64(Float64(-0.25 * Float64(Float64(20.0 * (Float64(c * a) ^ 4.0)) / Float64(a * (b ^ 7.0)))) - fma(a, Float64((c ^ 2.0) / (b ^ 3.0)), Float64(c / b))))
end

Wolfram

code[a_, b_, c_] := N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(N[(20.0 * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.6%

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

   1. *-commutative 14.6%

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

3. Simplified 14.6%

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

5. Taylor expanded in b around inf 97.5%

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

   1. fma-define 97.5%

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

   2. *-commutative 97.5%

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

   3. associate-+r+ 97.5%

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

   4. +-commutative 97.5%

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

   5. distribute-lft-out 97.5%

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

   6. mul-1-neg 97.5%

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

7. Simplified 97.5%

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

   1. pow1 97.5%

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

   2. associate-*r* 97.5%

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

   3. pow-prod-down 97.5%

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

9. Applied egg-rr 97.5%

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

    1. unpow1 97.5%

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

    2. *-commutative 97.5%

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

11. Simplified 97.5%

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

12. Final simplification 97.5%

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

Alternative 2: 97.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.6%

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

   1. *-commutative 14.6%

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

3. Simplified 14.6%

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

5. Taylor expanded in b around inf 96.8%

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

6. Final simplification 96.8%

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

Alternative 3: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.6%

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

   1. *-commutative 14.6%

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

3. Simplified 14.6%

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

5. Taylor expanded in b around inf 95.5%

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

   1. distribute-lft-out 95.5%

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

7. Simplified 95.5%

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

8. Final simplification 95.5%

   \[\leadsto \frac{-c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]
9. Add Preprocessing

Alternative 4: 90.4% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(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 14.6%

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

   1. *-commutative 14.6%

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

3. Simplified 14.6%

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

5. Taylor expanded in b around inf 92.3%

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

   1. mul-1-neg 92.3%

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

   2. distribute-neg-frac2 92.3%

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

7. Simplified 92.3%

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

8. Final simplification 92.3%

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

Alternative 5: 3.3% 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 14.6%

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

   1. *-commutative 14.6%

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

3. Simplified 14.6%

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

   1. add-sqr-sqrt 14.6%

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

   2. difference-of-squares 14.8%

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

   3. associate-*l* 14.8%

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

   4. sqrt-prod 14.8%

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

   5. metadata-eval 14.8%

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

   6. associate-*l* 14.8%

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

   7. sqrt-prod 14.8%

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

   8. metadata-eval 14.8%

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

6. Applied egg-rr 14.8%

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

   1. *-commutative 14.8%

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

   2. cancel-sign-sub-inv 14.8%

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

   3. metadata-eval 14.8%

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

8. Simplified 14.8%

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

9. Taylor expanded in b around inf 3.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.3%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}\right)}{a}} \]

    2. distribute-rgt-out 3.3%

       \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\sqrt{a \cdot c} \cdot \left(-2 + 2\right)\right)}}{a} \]

    3. metadata-eval 3.3%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{a \cdot c} \cdot \color{blue}{0}\right)}{a} \]

    4. mul0-rgt 3.3%

       \[\leadsto \frac{0.25 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.3%

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

11. Simplified 3.3%

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

12. Final simplification 3.3%

    \[\leadsto \frac{0}{a} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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)))