Quadratic roots, wide range

Average Error: 52.7 → 1.5

Time: 12.5s

Precision: binary64

Cost: 47168

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

Math

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

↓

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

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 b) (* a (/ (pow c 2.0) (pow b 3.0)))))
   (* -2.0 (* (pow c 3.0) (/ (pow a 2.0) (pow b 5.0)))))
  (* -0.25 (/ (* (pow (* c a) 4.0) 20.0) (* a (pow b 7.0))))))

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 (-((c / b) + (a * (pow(c, 2.0) / pow(b, 3.0)))) + (-2.0 * (pow(c, 3.0) * (pow(a, 2.0) / pow(b, 5.0))))) + (-0.25 * ((pow((c * a), 4.0) * 20.0) / (a * pow(b, 7.0))));
}

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 = (-((c / b) + (a * ((c ** 2.0d0) / (b ** 3.0d0)))) + ((-2.0d0) * ((c ** 3.0d0) * ((a ** 2.0d0) / (b ** 5.0d0))))) + ((-0.25d0) * ((((c * a) ** 4.0d0) * 20.0d0) / (a * (b ** 7.0d0))))
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 (-((c / b) + (a * (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) + (-2.0 * (Math.pow(c, 3.0) * (Math.pow(a, 2.0) / Math.pow(b, 5.0))))) + (-0.25 * ((Math.pow((c * a), 4.0) * 20.0) / (a * Math.pow(b, 7.0))));
}

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 (-((c / b) + (a * (math.pow(c, 2.0) / math.pow(b, 3.0)))) + (-2.0 * (math.pow(c, 3.0) * (math.pow(a, 2.0) / math.pow(b, 5.0))))) + (-0.25 * ((math.pow((c * a), 4.0) * 20.0) / (a * math.pow(b, 7.0))))

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(Float64(-Float64(Float64(c / b) + Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))))) + Float64(-2.0 * Float64((c ^ 3.0) * Float64((a ^ 2.0) / (b ^ 5.0))))) + Float64(-0.25 * Float64(Float64((Float64(c * a) ^ 4.0) * 20.0) / Float64(a * (b ^ 7.0)))))
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 = (-((c / b) + (a * ((c ^ 2.0) / (b ^ 3.0)))) + (-2.0 * ((c ^ 3.0) * ((a ^ 2.0) / (b ^ 5.0))))) + (-0.25 * ((((c * a) ^ 4.0) * 20.0) / (a * (b ^ 7.0))));
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[((-N[(N[(c / b), $MachinePrecision] + N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7

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

2. Simplified 52.7

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

   [Start] 52.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   rational.json-simplify-2 [=>] 52.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Taylor expanded in b around inf 1.5

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

4. Simplified 1.5

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

   [Start] 1.5	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
   rational.json-simplify-41 [=>] 1.5	\[ \color{blue}{-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
   rational.json-simplify-1 [<=] 1.5	\[ -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]
   rational.json-simplify-1 [=>] 1.5	\[ \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}} \]

5. Final simplification 1.5

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

Alternatives

Alternative 1
Error	1.8
Cost	41024

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

Alternative 2
Error	2.1
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 3
Error	2.2
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 4
Error	2.4
Cost	27456

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

Alternative 5
Error	2.3
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	5.6
Cost	14916

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 7
Error	3.0
Cost	13568

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

Alternative 8
Error	6.1
Cost	256

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

Reproduce

herbie shell --seed 2023067 
(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)))