Quadratic roots, wide range

Average Error: 82.22% → 0.13%

Time: 13.2s

Precision: binary64

Cost: 7488

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

Math

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

↓

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

FPCore

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

↓

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

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 0.5 * ((-4.0 * c) / (b + sqrt(((b * b) + (-4.0 * (c * 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

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.5d0 * (((-4.0d0) * c) / (b + sqrt(((b * b) + ((-4.0d0) * (c * 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);
}

↓

public static double code(double a, double b, double c) {
	return 0.5 * ((-4.0 * c) / (b + Math.sqrt(((b * b) + (-4.0 * (c * a))))));
}

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 0.5 * ((-4.0 * c) / (b + math.sqrt(((b * b) + (-4.0 * (c * 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

↓

function code(a, b, c)
	return Float64(0.5 * Float64(Float64(-4.0 * c) / Float64(b + sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a)))))))
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 = 0.5 * ((-4.0 * c) / (b + sqrt(((b * b) + (-4.0 * (c * 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]

↓

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

Derivation

1. Initial program 82.22

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

2. Simplified 82.22

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

   [Start] 82.22	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 82.22	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 81.85

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

4. Simplified 81.85

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

   [Start] 81.85	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 81.85	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 81.85	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   fma-def [<=] 81.85	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 81.85	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   fma-def [=>] 81.85	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]

5. Applied egg-rr 91.68

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}\right)} - 1} \]

6. Simplified 0.26

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

   [Start] 91.68	\[ e^{\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}\right)} - 1 \]
   expm1-def [=>] 88.12	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}\right)\right)} \]
   expm1-log1p [=>] 81.85	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}} \]
   *-lft-identity [<=] 81.85	\[ \frac{\color{blue}{1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \]
   *-commutative [=>] 81.85	\[ \frac{\color{blue}{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right) \cdot 1}}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \]
   associate-*r* [=>] 81.85	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right) \cdot 1}{\color{blue}{\left(\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a\right) \cdot 2}} \]
   *-commutative [<=] 81.85	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right) \cdot 1}{\color{blue}{\left(a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\right)} \cdot 2} \]
   times-frac [=>] 81.85	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)} \cdot \frac{1}{2}} \]
   metadata-eval [=>] 81.85	\[ \frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)} \cdot \color{blue}{0.5} \]
   *-commutative [=>] 81.85	\[ \color{blue}{0.5 \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}} \]
   associate-/r* [=>] 81.85	\[ 0.5 \cdot \color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{a}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]

7. Taylor expanded in c around 0 0.13

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

8. Applied egg-rr 0.13

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

9. Final simplification 0.13

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

Alternatives

Alternative 1
Error	4.52%
Cost	960

\[0.5 \cdot \frac{c \cdot 4}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)} \]

Alternative 2
Error	9.62%
Cost	256

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

Alternative 3
Error	98.29%
Cost	192

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

Reproduce

herbie shell --seed 2023089 
(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)))