Quadratic roots, medium range

Average Error: 43.5 → 0.2

Time: 16.3s

Precision: binary64

Cost: 7360

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

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

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 (* a -4.0)) (* 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 * (a * -4.0)) + (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 * (a * (-4.0d0))) + (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 * (a * -4.0)) + (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 * (a * -4.0)) + (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(c * Float64(a * -4.0)) + 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 * (a * -4.0)) + (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[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.5

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

2. Simplified 43.5

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

   [Start] 43.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   /-rgt-identity [<=] 43.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
   metadata-eval [<=] 43.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
   *-commutative [=>] 43.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
   associate-/l* [=>] 43.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
   associate-/l* [<=] 43.5	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
   associate-*r/ [<=] 43.5	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
   /-rgt-identity [<=] 43.5	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
   metadata-eval [<=] 43.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

3. Applied egg-rr 43.0

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

4. Simplified 42.9

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

   [Start] 43.0	\[ \frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) - \left(-b\right) \cdot \left(-b\right)}{\frac{a}{0.5} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - \left(-b\right)\right)} \]
   associate-/r* [=>] 43.0	\[ \color{blue}{\frac{\frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) - \left(-b\right) \cdot \left(-b\right)}{\frac{a}{0.5}}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - \left(-b\right)}} \]
   fma-def [<=] 42.9	\[ \frac{\frac{\color{blue}{\left(b \cdot b + \left(a \cdot c\right) \cdot -4\right)} - \left(-b\right) \cdot \left(-b\right)}{\frac{a}{0.5}}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - \left(-b\right)} \]
   +-commutative [=>] 42.9	\[ \frac{\frac{\color{blue}{\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)} - \left(-b\right) \cdot \left(-b\right)}{\frac{a}{0.5}}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - \left(-b\right)} \]
   *-commutative [=>] 42.9	\[ \frac{\frac{\left(\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b\right) - \left(-b\right) \cdot \left(-b\right)}{\frac{a}{0.5}}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - \left(-b\right)} \]
   fma-def [=>] 42.9	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - \left(-b\right) \cdot \left(-b\right)}{\frac{a}{0.5}}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - \left(-b\right)} \]
   sqr-neg [=>] 42.9	\[ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right) - \color{blue}{b \cdot b}}{\frac{a}{0.5}}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - \left(-b\right)} \]
   fma-def [<=] 42.9	\[ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right) - b \cdot b}{\frac{a}{0.5}}}{\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - \left(-b\right)} \]
   +-commutative [=>] 42.9	\[ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right) - b \cdot b}{\frac{a}{0.5}}}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} - \left(-b\right)} \]
   *-commutative [=>] 42.9	\[ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right) - b \cdot b}{\frac{a}{0.5}}}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} - \left(-b\right)} \]
   fma-def [=>] 42.9	\[ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right) - b \cdot b}{\frac{a}{0.5}}}{\sqrt{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}} - \left(-b\right)} \]

5. Taylor expanded in c around 0 0.2

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

6. Simplified 0.2

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

   [Start] 0.2	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{\frac{a}{0.5}}}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - \left(-b\right)} \]
   *-commutative [=>] 0.2	\[ \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot -4}}{\frac{a}{0.5}}}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - \left(-b\right)} \]
   associate-*r* [<=] 0.2	\[ \frac{\frac{\color{blue}{c \cdot \left(a \cdot -4\right)}}{\frac{a}{0.5}}}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - \left(-b\right)} \]
   *-commutative [=>] 0.2	\[ \frac{\frac{c \cdot \color{blue}{\left(-4 \cdot a\right)}}{\frac{a}{0.5}}}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - \left(-b\right)} \]

7. Applied egg-rr 0.2

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

8. Taylor expanded in c around 0 0.2

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

9. Simplified 0.2

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

   [Start] 0.2	\[ \frac{-2 \cdot c}{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b} - \left(-b\right)} \]
   *-commutative [=>] 0.2	\[ \frac{\color{blue}{c \cdot -2}}{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b} - \left(-b\right)} \]

10. Final simplification 0.2

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

Alternatives

Alternative 1
Error	5.8
Cost	1344

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

Alternative 2
Error	5.9
Cost	832

\[\frac{0.5}{0.5 \cdot \frac{a}{b} + -0.5 \cdot \frac{b}{c}} \]

Alternative 3
Error	12.2
Cost	256

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

Alternative 4
Error	63.0
Cost	192

\[\frac{b}{a} \]

Reproduce

herbie shell --seed 2023011 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))