Quadratic roots, medium range

Average Error: 43.7 → 3.1

Time: 13.2s

Precision: binary64

Cost: 67136

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{{c}^{4}}{{b}^{6}}\\ -1 \cdot \frac{c}{b} + \left(\left(-2 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{5}}\right) + -0.25 \cdot \left(\left(16 \cdot t_0 + 4 \cdot t_0\right) \cdot \frac{{a}^{3}}{b}\right)\right) + -1 \cdot \left(a \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (pow c 4.0) (pow b 6.0))))
   (+
    (* -1.0 (/ c b))
    (+
     (+
      (* -2.0 (* (pow a 2.0) (/ (pow c 3.0) (pow b 5.0))))
      (* -0.25 (* (+ (* 16.0 t_0) (* 4.0 t_0)) (/ (pow a 3.0) b))))
     (* -1.0 (* a (/ (pow c 2.0) (pow b 3.0))))))))

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) {
	double t_0 = pow(c, 4.0) / pow(b, 6.0);
	return (-1.0 * (c / b)) + (((-2.0 * (pow(a, 2.0) * (pow(c, 3.0) / pow(b, 5.0)))) + (-0.25 * (((16.0 * t_0) + (4.0 * t_0)) * (pow(a, 3.0) / b)))) + (-1.0 * (a * (pow(c, 2.0) / pow(b, 3.0)))));
}

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
    real(8) :: t_0
    t_0 = (c ** 4.0d0) / (b ** 6.0d0)
    code = ((-1.0d0) * (c / b)) + ((((-2.0d0) * ((a ** 2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0)))) + ((-0.25d0) * (((16.0d0 * t_0) + (4.0d0 * t_0)) * ((a ** 3.0d0) / b)))) + ((-1.0d0) * (a * ((c ** 2.0d0) / (b ** 3.0d0)))))
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) {
	double t_0 = Math.pow(c, 4.0) / Math.pow(b, 6.0);
	return (-1.0 * (c / b)) + (((-2.0 * (Math.pow(a, 2.0) * (Math.pow(c, 3.0) / Math.pow(b, 5.0)))) + (-0.25 * (((16.0 * t_0) + (4.0 * t_0)) * (Math.pow(a, 3.0) / b)))) + (-1.0 * (a * (Math.pow(c, 2.0) / Math.pow(b, 3.0)))));
}

Python

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

↓

def code(a, b, c):
	t_0 = math.pow(c, 4.0) / math.pow(b, 6.0)
	return (-1.0 * (c / b)) + (((-2.0 * (math.pow(a, 2.0) * (math.pow(c, 3.0) / math.pow(b, 5.0)))) + (-0.25 * (((16.0 * t_0) + (4.0 * t_0)) * (math.pow(a, 3.0) / b)))) + (-1.0 * (a * (math.pow(c, 2.0) / math.pow(b, 3.0)))))

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)
	t_0 = Float64((c ^ 4.0) / (b ^ 6.0))
	return Float64(Float64(-1.0 * Float64(c / b)) + Float64(Float64(Float64(-2.0 * Float64((a ^ 2.0) * Float64((c ^ 3.0) / (b ^ 5.0)))) + Float64(-0.25 * Float64(Float64(Float64(16.0 * t_0) + Float64(4.0 * t_0)) * Float64((a ^ 3.0) / b)))) + Float64(-1.0 * Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))))))
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)
	t_0 = (c ^ 4.0) / (b ^ 6.0);
	tmp = (-1.0 * (c / b)) + (((-2.0 * ((a ^ 2.0) * ((c ^ 3.0) / (b ^ 5.0)))) + (-0.25 * (((16.0 * t_0) + (4.0 * t_0)) * ((a ^ 3.0) / b)))) + (-1.0 * (a * ((c ^ 2.0) / (b ^ 3.0)))));
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_] := Block[{t$95$0 = N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[(16.0 * t$95$0), $MachinePrecision] + N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $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 3.1

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

4. Simplified 3.1

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

   [Start] 3.1	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
   rational.json-simplify-41 [=>] 3.1	\[ \color{blue}{-1 \cdot \frac{c}{b} + \left(\left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

5. Taylor expanded in c around 0 3.1

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

6. Final simplification 3.1

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

Alternatives

Alternative 1
Error	3.1
Cost	47168

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

Alternative 2
Error	3.3
Cost	41024

\[\frac{-4 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}} + \left(-2 \cdot \left(c \cdot \frac{a}{b} + \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}\right) + -0.5 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}\right)}{a \cdot 2} \]

Alternative 3
Error	4.1
Cost	33472

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

Alternative 4
Error	4.2
Cost	27584

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

Alternative 5
Error	4.3
Cost	27456

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

Alternative 6
Error	10.2
Cost	14916

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 7
Error	5.8
Cost	13700

\[\begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{c}{b} + a \cdot \frac{{c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 8
Error	11.2
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \leq 0.000425:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 9
Error	12.1
Cost	256

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

Alternative 10
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2023068 
(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)))