Quadratic roots, medium range

Average Error: 43.8 → 0.2

Time: 15.8s

Precision: binary64

Cost: 13760

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

Math

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

↓

\[\frac{-4 \cdot c}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{0.5}} \]

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) (/ (+ b (sqrt (fma b b (* c (* -4.0 a))))) 0.5)))

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) / ((b + sqrt(fma(b, b, (c * (-4.0 * a))))) / 0.5);
}

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(-4.0 * c) / Float64(Float64(b + sqrt(fma(b, b, Float64(c * Float64(-4.0 * a))))) / 0.5))
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[(-4.0 * c), $MachinePrecision] / N[(N[(b + N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.8

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

2. Simplified 43.7

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   Proof

   [Start] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   /-rgt-identity [<=] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
   metadata-eval [<=] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
   *-commutative [=>] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
   associate-/l* [=>] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
   associate-/l* [<=] 43.8	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
   associate-*r/ [<=] 43.8	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
   /-rgt-identity [<=] 43.8	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
   metadata-eval [<=] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

3. Applied egg-rr 43.2

   \[\leadsto \color{blue}{\frac{0.5}{\frac{a \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}\right)}{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) - b \cdot b}}} \]

4. Applied egg-rr 49.3

   \[\leadsto \color{blue}{\log \left({\left(e^{0.5}\right)}^{\left(\frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) - b \cdot b}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}\right)}\right)}\right)} \]

5. Simplified 0.2

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]
   Proof

   [Start] 49.3	\[ \log \left({\left(e^{0.5}\right)}^{\left(\frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) - b \cdot b}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}\right)}\right)}\right) \]
   log-pow [=>] 43.2	\[ \color{blue}{\frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) - b \cdot b}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}\right)} \cdot \log \left(e^{0.5}\right)} \]
   associate-/r* [=>] 43.2	\[ \color{blue}{\frac{\frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) - b \cdot b}{a}}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}} \cdot \log \left(e^{0.5}\right) \]
   associate-*l/ [=>] 43.2	\[ \color{blue}{\frac{\frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) - b \cdot b}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}} \]
   fma-udef [=>] 43.1	\[ \frac{\frac{\color{blue}{\left(b \cdot b + \left(a \cdot c\right) \cdot -4\right)} - b \cdot b}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}} \]
   +-commutative [<=] 43.1	\[ \frac{\frac{\color{blue}{\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)} - b \cdot b}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}} \]
   associate-+r- [<=] 0.2	\[ \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot -4 + \left(b \cdot b - b \cdot b\right)}}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}} \]
   *-commutative [<=] 0.2	\[ \frac{\frac{\color{blue}{\left(c \cdot a\right)} \cdot -4 + \left(b \cdot b - b \cdot b\right)}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}} \]
   fma-def [=>] 0.2	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, b \cdot b - b \cdot b\right)}}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}} \]
   +-inverses [=>] 0.2	\[ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, \color{blue}{0}\right)}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}} \]

6. Applied egg-rr 39.9

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)} - 1} \]

7. Simplified 0.2

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{0.5}}} \]
   Proof

   [Start] 39.9	\[ e^{\mathsf{log1p}\left(\frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)} - 1 \]
   expm1-def [=>] 10.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)\right)} \]
   expm1-log1p [=>] 0.2	\[ \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a} \cdot \log \left(e^{0.5}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]
   associate-/l* [=>] 0.2	\[ \color{blue}{\frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{\log \left(e^{0.5}\right)}}} \]
   *-commutative [=>] 0.2	\[ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot -4}\right)}}{\log \left(e^{0.5}\right)}} \]
   associate-*r* [<=] 0.2	\[ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(a \cdot -4\right)}\right)}}{\log \left(e^{0.5}\right)}} \]
   *-commutative [=>] 0.2	\[ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(-4 \cdot a\right)}\right)}}{\log \left(e^{0.5}\right)}} \]
   rem-log-exp [=>] 0.2	\[ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{\color{blue}{0.5}}} \]

8. Taylor expanded in c around 0 0.2

   \[\leadsto \frac{\color{blue}{-4 \cdot c}}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{0.5}} \]

9. Final simplification 0.2

   \[\leadsto \frac{-4 \cdot c}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{0.5}} \]

Alternatives

Alternative 1
Error	5.8
Cost	15684

\[\begin{array}{l} t_0 := b \cdot b + c \cdot \left(-4 \cdot a\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -300:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a}}{\frac{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}{0.5}}\\ \end{array} \]

Alternative 2
Error	4.1
Cost	8320

\[\frac{\frac{0.5}{a}}{\frac{c \cdot a + \left(c \cdot a\right) \cdot -0.5}{{b}^{3}} + \left(-0.5 \cdot \frac{b}{c \cdot a} + 0.5 \cdot \frac{1}{b}\right)} \]

Alternative 3
Error	5.9
Cost	7744

\[\frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{a}}{\frac{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}{0.5}} \]

Alternative 4
Error	5.9
Cost	7104

\[\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \frac{-0.5}{\frac{c}{b}}\right)} \]

Alternative 5
Error	6.0
Cost	832

\[\frac{0.5}{0.5 \cdot \frac{a}{b} + -0.5 \cdot \frac{b}{c}} \]

Alternative 6
Error	12.1
Cost	256

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

Reproduce

herbie shell --seed 2023031 
(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)))