Quadratic roots, medium range

Average Error: 43.7 → 0.4

Time: 15.1s

Precision: binary64

Cost: 14016

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

Math

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

↓

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

FPCore

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

↓

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

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 ((-4.0 * (c * a)) / (b + sqrt(fma(b, b, (c * (-4.0 * a)))))) / (a * 2.0);
}

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(-4.0 * Float64(c * a)) / Float64(b + sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))))) / Float64(a * 2.0))
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[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $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
   (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= *-commutative_binary64 (*.f64 2 a))): 0 points increase in error, 2 points decrease in error

3. Applied egg-rr 43.2

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

4. Taylor expanded in b around 0 0.4

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

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	6.7
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.004:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{\frac{a \cdot \left(c \cdot c\right)}{b}}{b \cdot b}\\ \end{array} \]

Alternative 2
Error	4.1
Cost	8448

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

Alternative 3
Error	4.1
Cost	8320

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

Alternative 4
Error	6.1
Cost	1344

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

Alternative 5
Error	6.1
Cost	1344

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

Alternative 6
Error	6.1
Cost	1024

\[\frac{-c}{b} - \frac{\frac{a \cdot \left(c \cdot c\right)}{b}}{b \cdot b} \]

Alternative 7
Error	12.1
Cost	256

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

Alternative 8
Error	63.0
Cost	192

\[\frac{b}{a} \]

Reproduce

herbie shell --seed 2022349 
(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)))