Quadratic roots, medium range

Average Error: 43.7 → 3.0

Time: 4.3s

Precision: binary64

Cost: 53824

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

Math

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

↓

\[-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{20 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}}}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (+
  (* -1.0 (/ (* (pow c 2.0) a) (pow b 3.0)))
  (+
   (* -1.0 (/ c b))
   (+
    (* -0.25 (/ (* 20.0 (/ (* (pow c 4.0) (pow a 3.0)) (pow b 6.0))) b))
    (* -2.0 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))))))

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 (-1.0 * ((pow(c, 2.0) * a) / pow(b, 3.0))) + ((-1.0 * (c / b)) + ((-0.25 * ((20.0 * ((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 6.0))) / b)) + (-2.0 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-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 = ((-1.0d0) * (((c ** 2.0d0) * a) / (b ** 3.0d0))) + (((-1.0d0) * (c / b)) + (((-0.25d0) * ((20.0d0 * (((c ** 4.0d0) * (a ** 3.0d0)) / (b ** 6.0d0))) / b)) + ((-2.0d0) * (((c ** 3.0d0) * (a ** 2.0d0)) / (b ** 5.0d0)))))
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 (-1.0 * ((Math.pow(c, 2.0) * a) / Math.pow(b, 3.0))) + ((-1.0 * (c / b)) + ((-0.25 * ((20.0 * ((Math.pow(c, 4.0) * Math.pow(a, 3.0)) / Math.pow(b, 6.0))) / b)) + (-2.0 * ((Math.pow(c, 3.0) * Math.pow(a, 2.0)) / Math.pow(b, 5.0)))));
}

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 (-1.0 * ((math.pow(c, 2.0) * a) / math.pow(b, 3.0))) + ((-1.0 * (c / b)) + ((-0.25 * ((20.0 * ((math.pow(c, 4.0) * math.pow(a, 3.0)) / math.pow(b, 6.0))) / b)) + (-2.0 * ((math.pow(c, 3.0) * math.pow(a, 2.0)) / math.pow(b, 5.0)))))

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(-1.0 * Float64(Float64((c ^ 2.0) * a) / (b ^ 3.0))) + Float64(Float64(-1.0 * Float64(c / b)) + Float64(Float64(-0.25 * Float64(Float64(20.0 * Float64(Float64((c ^ 4.0) * (a ^ 3.0)) / (b ^ 6.0))) / b)) + Float64(-2.0 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))))))
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 = (-1.0 * (((c ^ 2.0) * a) / (b ^ 3.0))) + ((-1.0 * (c / b)) + ((-0.25 * ((20.0 * (((c ^ 4.0) * (a ^ 3.0)) / (b ^ 6.0))) / b)) + (-2.0 * (((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0)))));
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[(-1.0 * N[(N[(N[Power[c, 2.0], $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(N[(20.0 * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.7

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

2. Simplified 43.7

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

   [Start] 43.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   rational.json-simplify-2 [=>] 43.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Taylor expanded in a around 0 3.0

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

4. Taylor expanded in c around 0 3.0

   \[\leadsto -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{\color{blue}{20 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}}}}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]

5. Final simplification 3.0

   \[\leadsto -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{20 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}}}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]

Alternatives

Alternative 1
Error	3.0
Cost	53696

\[-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \left(20 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\right) + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]

Alternative 2
Error	4.0
Cost	33664

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

Alternative 3
Error	10.1
Cost	14916

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	6.0
Cost	13760

\[-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b} \]

Alternative 5
Error	12.1
Cost	320

\[-1 \cdot \frac{c}{b} \]

Reproduce

herbie shell --seed 2023053 
(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)))