Quadratic roots, full range

Average Accuracy: 46.9% → 87.2%

Time: 18.9s

Precision: binary64

Cost: 13964

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -6.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1900000000000:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+108)
   (- (/ c b) (/ b a))
   (if (<= b -6.8e-221)
     (/ (- (sqrt (+ (* b b) (* c (* a -4.0)))) b) (* a 2.0))
     (if (<= b 1900000000000.0)
       (/ (* c -2.0) (+ b (hypot b (sqrt (* a (* c -4.0))))))
       (/ (- c) b)))))

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) {
	double tmp;
	if (b <= -5e+108) {
		tmp = (c / b) - (b / a);
	} else if (b <= -6.8e-221) {
		tmp = (sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0);
	} else if (b <= 1900000000000.0) {
		tmp = (c * -2.0) / (b + hypot(b, sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

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) {
	double tmp;
	if (b <= -5e+108) {
		tmp = (c / b) - (b / a);
	} else if (b <= -6.8e-221) {
		tmp = (Math.sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0);
	} else if (b <= 1900000000000.0) {
		tmp = (c * -2.0) / (b + Math.hypot(b, Math.sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

↓

def code(a, b, c):
	tmp = 0
	if b <= -5e+108:
		tmp = (c / b) - (b / a)
	elif b <= -6.8e-221:
		tmp = (math.sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0)
	elif b <= 1900000000000.0:
		tmp = (c * -2.0) / (b + math.hypot(b, math.sqrt((a * (c * -4.0)))))
	else:
		tmp = -c / b
	return tmp

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)
	tmp = 0.0
	if (b <= -5e+108)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -6.8e-221)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	elseif (b <= 1900000000000.0)
		tmp = Float64(Float64(c * -2.0) / Float64(b + hypot(b, sqrt(Float64(a * Float64(c * -4.0))))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+108)
		tmp = (c / b) - (b / a);
	elseif (b <= -6.8e-221)
		tmp = (sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0);
	elseif (b <= 1900000000000.0)
		tmp = (c * -2.0) / (b + hypot(b, sqrt((a * (c * -4.0)))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
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_] := If[LessEqual[b, -5e+108], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.8e-221], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1900000000000.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.99999999999999991e108

   1. Initial program 23.9%

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

   2. Simplified 23.9%

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

      [Start] 23.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 23.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 94.4%

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

   4. Simplified 94.4%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
      Proof

      [Start] 94.4	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 94.4	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 94.4	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -4.99999999999999991e108 < b < -6.8000000000000003e-221

   1. Initial program 88.0%

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

   if -6.8000000000000003e-221 < b < 1.9e12

   1. Initial program 62.3%

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

   2. Simplified 62.3%

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

      [Start] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 62.3	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 62.3	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 62.3	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Applied egg-rr 61.3%

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

   4. Taylor expanded in a around 0 77.1%

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

   5. Simplified 77.1%

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

      [Start] 77.1	\[ \frac{-2 \cdot c}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]
      *-commutative [=>] 77.1	\[ \frac{\color{blue}{c \cdot -2}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]

   if 1.9e12 < b

   1. Initial program 12.6%

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

   2. Simplified 12.6%

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

      [Start] 12.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 12.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 12.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 12.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 12.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 12.6	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 12.6	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 12.6	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 12.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Taylor expanded in b around inf 91.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   4. Simplified 91.7%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]
      Proof

      [Start] 91.7	\[ -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 91.7	\[ \color{blue}{-\frac{c}{b}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -6.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1900000000000:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.2%
Cost	13964

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-297}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2100000000000:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Accuracy	87.2%
Cost	13964

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;b \leq -1.75 \cdot 10^{+103}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 680000000000:\\ \;\;\;\;\frac{-2}{\frac{b + \mathsf{hypot}\left(b, \sqrt{t_0}\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Accuracy	83.6%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+103}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Accuracy	83.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -4.65 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Accuracy	79.1%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Accuracy	79.1%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Accuracy	64.1%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-171}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8
Accuracy	37.4%
Cost	256

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

Alternative 9
Accuracy	11.2%
Cost	192

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

Reproduce

herbie shell --seed 2023129 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))