Quadratic roots, narrow range

Average Error: 28.7 → 0.3

Time: 14.8s

Precision: binary64

Cost: 7360

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double a, double b, double c) {
	return (c * -2.0) / (b + sqrt((((c * -4.0) * a) + (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 = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

↓

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

↓

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

Python

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

↓

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

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

↓

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

MATLAB

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

↓

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

↓

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

Derivation

1. Initial program 28.7

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

2. Simplified 28.7

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

   [Start] 28.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 28.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 27.9

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

4. Simplified 27.9

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

   [Start] 27.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 27.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 27.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   fma-def [<=] 27.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 27.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   fma-def [=>] 27.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]

5. Applied egg-rr 27.9

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

6. Simplified 27.9

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

   [Start] 27.9	\[ \frac{-\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{1}{a \cdot -2} \]
   associate-*l/ [=>] 27.9	\[ \color{blue}{\frac{\left(-\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]
   *-commutative [=>] 27.9	\[ \frac{\left(-\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, \color{blue}{-4 \cdot a}, b \cdot b\right)}} \]

7. Taylor expanded in b around 0 0.3

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

8. Simplified 0.3

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

   [Start] 0.3	\[ \frac{-2 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   *-commutative [=>] 0.3	\[ \frac{\color{blue}{c \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]

9. Taylor expanded in c around 0 0.3

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

10. Simplified 0.3

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

    [Start] 0.3	\[ \frac{c \cdot -2}{b + \sqrt{{b}^{2} + -4 \cdot \left(c \cdot a\right)}} \]
    +-commutative [=>] 0.3	\[ \frac{c \cdot -2}{b + \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right) + {b}^{2}}}} \]
    unpow2 [=>] 0.3	\[ \frac{c \cdot -2}{b + \sqrt{-4 \cdot \left(c \cdot a\right) + \color{blue}{b \cdot b}}} \]
    associate-*r* [=>] 0.3	\[ \frac{c \cdot -2}{b + \sqrt{\color{blue}{\left(-4 \cdot c\right) \cdot a} + b \cdot b}} \]

11. Final simplification 0.3

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

Alternatives

Alternative 1
Error	9.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 31.5:\\ \;\;\;\;\left(\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}\\ \end{array} \]

Alternative 2
Error	11.4
Cost	960

\[-2 \cdot \frac{1}{2 \cdot \frac{b}{c} + -2 \cdot \frac{a}{b}} \]

Alternative 3
Error	11.4
Cost	960

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

Alternative 4
Error	11.3
Cost	960

\[\frac{c \cdot -2}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2} \]

Alternative 5
Error	22.8
Cost	256

\[-\frac{c}{b} \]

Reproduce

herbie shell --seed 2023053 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))