Quadratic roots, wide range

Percentage Accurate: 17.6% → 99.2%

Time: 10.7s

Alternatives: 3

Speedup: 3.6×

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: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{a}{\frac{-1}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}{4 \cdot \left(a \cdot c\right)}}}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.5
  (/ a (/ -1.0 (/ (+ (sqrt (fma (* -4.0 c) a (* b b))) b) (* 4.0 (* a c)))))))

C

double code(double a, double b, double c) {
	return 0.5 / (a / (-1.0 / ((sqrt(fma((-4.0 * c), a, (b * b))) + b) / (4.0 * (a * c)))));
}

Julia

function code(a, b, c)
	return Float64(0.5 / Float64(a / Float64(-1.0 / Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b) / Float64(4.0 * Float64(a * c))))))
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(a / N[(-1.0 / N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.1%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 18.1

      \[\leadsto \frac{0.5}{\color{blue}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}} \]

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 18.1

       \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}} \]

4. Applied rewrites 18.1%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. sub-div N/A

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

   9. lower--.f64 N/A

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

6. Applied rewrites 18.5%

   \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b} - \frac{b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}}} \]

8. Applied rewrites 19.7%

   \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{\frac{1}{\frac{-\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{\mathsf{fma}\left(b, b, -\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}}}}} \]

9. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 99.2

       \[\leadsto \frac{0.5}{\frac{a}{\frac{1}{\frac{-\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{4 \cdot \color{blue}{\left(a \cdot c\right)}}}}} \]

11. Applied rewrites 99.2%

    \[\leadsto \frac{0.5}{\frac{a}{\frac{1}{\frac{-\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{\color{blue}{4 \cdot \left(a \cdot c\right)}}}}} \]

12. Final simplification 99.2%

    \[\leadsto \frac{0.5}{\frac{a}{\frac{-1}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}{4 \cdot \left(a \cdot c\right)}}}} \]
13. Add Preprocessing

Alternative 2: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 97.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 95.3%

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

12. Final simplification 95.3%

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

Alternative 3: 90.5% accurate, 3.6× 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 18.1%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

   4. lower-neg.f64 90.6

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

5. Applied rewrites 90.6%

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

Reproduce

herbie shell --seed 2024318 
(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)))