Quadratic roots, narrow range

Average Error: 29.0 → 5.5

Time: 45.0s

Precision: binary64

Cost: 47168

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (+
  (- (+ (/ 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(c / b) + Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))) + Float64(Float64(-2.0 * Float64(Float64((c ^ 3.0) * (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[(c / b), $MachinePrecision] + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(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] + 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]), $MachinePrecision]

Derivation

1. Initial program 29.0

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

2. Simplified 29.0

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

   [Start] 29.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   rational.json-simplify-2 [=>] 29.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Taylor expanded in b around inf 5.5

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

4. Simplified 5.5

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

   [Start] 5.5	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
   rational.json-simplify-41 [=>] 5.5	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)} \]
   rational.json-simplify-41 [=>] 5.4	\[ \color{blue}{-1 \cdot \frac{c}{b} + \left(\left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
   rational.json-simplify-41 [=>] 5.5	\[ \color{blue}{\left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) + \left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}\right)} \]
   rational.json-simplify-1 [=>] 5.5	\[ \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)} \]

5. Final simplification 5.5

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

Alternatives

Alternative 1
Error	5.6
Cost	41664

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \frac{0.5 \cdot \frac{t_0}{b} + \left(\left(-0.125 \cdot \frac{16 \cdot {\left(c \cdot a\right)}^{2}}{{b}^{3}} + 0.0625 \cdot \frac{-64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{5}}\right) + -0.5 \cdot \frac{{t_0}^{4} \cdot 0.078125}{{b}^{7}}\right)}{a \cdot 2} \end{array} \]

Alternative 2
Error	5.6
Cost	41024

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

Alternative 3
Error	6.8
Cost	33796

\[\begin{array}{l} \mathbf{if}\;b \leq 0.00047:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left({b}^{2} + \left(c \cdot a\right) \cdot -8\right) - \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b} + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 4
Error	6.8
Cost	33604

\[\begin{array}{l} \mathbf{if}\;b \leq 0.00046:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left({b}^{2} + \left(c \cdot a\right) \cdot -8\right) - \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\\ \end{array} \]

Alternative 5
Error	9.3
Cost	29316

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.008:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 6
Error	6.9
Cost	28100

\[\begin{array}{l} \mathbf{if}\;b \leq 0.00045:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left({b}^{2} + \left(c \cdot a\right) \cdot -8\right) - \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{c \cdot \left(a \cdot -4\right)}{b} + \left(-0.125 \cdot \frac{16 \cdot {\left(c \cdot a\right)}^{2}}{{b}^{3}} + 0.0625 \cdot \frac{-64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)}{a \cdot 2}\\ \end{array} \]

Alternative 7
Error	7.0
Cost	27588

\[\begin{array}{l} \mathbf{if}\;b \leq 0.00047:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left({b}^{2} + \left(c \cdot a\right) \cdot -8\right) - \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{c \cdot a}{b} + \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}\right) + -4 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}}{a \cdot 2}\\ \end{array} \]

Alternative 8
Error	9.3
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.008:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 9
Error	15.3
Cost	14852

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 10
Error	22.4
Cost	256

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

Reproduce

herbie shell --seed 2023077 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))