Quadratic roots, wide range

Average Error: 52.7 → 1.5

Time: 16.8s

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

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

↓

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

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 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_best-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

   \[\leadsto \color{blue}{\left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot {\left(c \cdot a\right)}^{2}\right)}^{2} + 16 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}\right)} \]
   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_best-simplify-1 [=>] 1.5	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \color{blue}{\left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -1 \cdot \frac{c}{b}\right)}\right) \]
   rational_best-simplify-43 [=>] 1.5	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)} \]
   rational_best-simplify-43 [=>] 1.5	\[ \color{blue}{\left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) + \left(-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
   rational_best-simplify-1 [<=] 1.5	\[ \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) + \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}\right)} \]

5. Taylor expanded in c around 0 1.5

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

6. Simplified 1.5

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

   [Start] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}\right) \]
   exponential-simplify-27 [=>] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{4 \cdot \color{blue}{{\left(a \cdot c\right)}^{4}} + 16 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}\right) \]
   rational_best-simplify-2 [<=] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{4 \cdot {\color{blue}{\left(c \cdot a\right)}}^{4} + 16 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}\right) \]

7. Taylor expanded in c around 0 1.5

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

8. Simplified 1.5

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

   [Start] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{c}^{4} \cdot \left(16 \cdot {a}^{4} + 4 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
   rational_best-simplify-47 [<=] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{{c}^{4} \cdot \left(4 \cdot {a}^{4}\right) + {c}^{4} \cdot \left(16 \cdot {a}^{4}\right)}}{a \cdot {b}^{7}}\right) \]
   rational_best-simplify-44 [<=] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{4 \cdot \left({c}^{4} \cdot {a}^{4}\right)} + {c}^{4} \cdot \left(16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
   rational_best-simplify-44 [<=] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + \color{blue}{16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}}{a \cdot {b}^{7}}\right) \]
   rational_best-simplify-2 [=>] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot 4} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
   rational_best-simplify-2 [=>] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 4 + \color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot 16}}{a \cdot {b}^{7}}\right) \]
   rational_best-simplify-47 [=>] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(16 + 4\right)}}{a \cdot {b}^{7}}\right) \]
   exponential-simplify-27 [=>] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \left(16 + 4\right)}{a \cdot {b}^{7}}\right) \]
   rational_best-simplify-2 [<=] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\color{blue}{\left(c \cdot a\right)}}^{4} \cdot \left(16 + 4\right)}{a \cdot {b}^{7}}\right) \]
   metadata-eval [=>] 1.5	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot \color{blue}{20}}{a \cdot {b}^{7}}\right) \]

9. Final simplification 1.5

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

Alternatives

Alternative 1
Error	2.0
Cost	33472

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

Alternative 2
Error	2.3
Cost	27584

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

Alternative 3
Error	2.3
Cost	27456

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

Alternative 4
Error	3.0
Cost	13568

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

Alternative 5
Error	6.0
Cost	256

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

Reproduce

herbie shell --seed 2023092 
(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)))