Quadratic roots, medium range

?

Percentage Accurate: 31.6% → 95.4%
Time: 10.1s
Precision: binary64
Cost: 47040

?

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-5 \cdot \frac{{c}^{4}}{\frac{{b}^{6}}{{a}^{3}}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (-
   (fma
    -2.0
    (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
    (/ (* -5.0 (/ (pow c 4.0) (/ (pow b 6.0) (pow a 3.0)))) b))
   (/ c b))
  (/ (* c c) (/ (pow b 3.0) a))))
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 (fma(-2.0, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), ((-5.0 * (pow(c, 4.0) / (pow(b, 6.0) / pow(a, 3.0)))) / b)) - (c / b)) - ((c * c) / (pow(b, 3.0) / a));
}

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(fma(-2.0, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), Float64(Float64(-5.0 * Float64((c ^ 4.0) / Float64((b ^ 6.0) / (a ^ 3.0)))) / b)) - Float64(c / b)) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)))
end

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[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-5.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[(N[Power[b, 6.0], $MachinePrecision] / N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-5 \cdot \frac{{c}^{4}}{\frac{{b}^{6}}{{a}^{3}}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 7 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

  1. Initial program 30.4%

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

  2. Simplified30.5%

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

    [Start]30.4%

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

    /-rgt-identity [<=]30.4%

    \[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

    metadata-eval [<=]30.4%

    \[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

    associate-/l* [<=]30.4%

    \[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]

    associate-*r/ [<=]30.4%

    \[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]

    +-commutative [=>]30.4%

    \[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]

    unsub-neg [=>]30.4%

    \[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]

    fma-neg [=>]30.5%

    \[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]

    associate-*l* [=>]30.5%

    \[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

    *-commutative [=>]30.5%

    \[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

    distribute-rgt-neg-in [=>]30.5%

    \[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

    metadata-eval [=>]30.5%

    \[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]

    associate-/r* [=>]30.5%

    \[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]

    metadata-eval [=>]30.5%

    \[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]

    metadata-eval [=>]30.5%

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

  3. Taylor expanded in a around 0 94.3%

    \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

  4. Simplified94.3%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{\left(-0.25 \cdot {a}^{3}\right) \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
    Step-by-step derivation

    [Start]94.3%

    \[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]

    +-commutative [=>]94.3%

    \[ \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

    mul-1-neg [=>]94.3%

    \[ \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

    unsub-neg [=>]94.3%

    \[ \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

  5. Taylor expanded in c around 0 94.3%

    \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{\color{blue}{-5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

  6. Simplified94.3%

    \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{\color{blue}{-5 \cdot \frac{{c}^{4}}{\frac{{b}^{6}}{{a}^{3}}}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]
    Step-by-step derivation

    [Start]94.3%

    \[ \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

    associate-/l* [=>]94.3%

    \[ \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-5 \cdot \color{blue}{\frac{{c}^{4}}{\frac{{b}^{6}}{{a}^{3}}}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

  7. Final simplification94.3%

    \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-5 \cdot \frac{{c}^{4}}{\frac{{b}^{6}}{{a}^{3}}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

Alternatives

Alternative 1
Accuracy95.4%
Cost47040
\[\left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-5 \cdot \frac{{c}^{4}}{\frac{{b}^{6}}{{a}^{3}}}}{b}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]
Alternative 2
Accuracy93.8%
Cost20736
\[-2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \left(\frac{c}{b} + \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) \]
Alternative 3
Accuracy90.8%
Cost15684
\[\begin{array}{l} t_0 := b \cdot b - c \cdot \left(a \cdot 4\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -0.9:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + t_1} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]
Alternative 4
Accuracy90.8%
Cost15684
\[\begin{array}{l} t_0 := b \cdot b - c \cdot \left(a \cdot 4\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -0.9:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]
Alternative 5
Accuracy90.6%
Cost14788
\[\begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\\ \mathbf{if}\;\frac{t_0}{a \cdot 2} \leq -0.9:\\ \;\;\;\;\frac{0.5}{a} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]
Alternative 6
Accuracy90.6%
Cost7232
\[\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]
Alternative 7
Accuracy81.1%
Cost256
\[\frac{-c}{b} \]

Reproduce?

herbie shell --seed 2023171 
(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)))