quadp (p42, positive)

Average Error: 34.3 → 8.2

Time: 13.4s

Precision: binary64

Cost: 13964

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.161151587419122 \cdot 10^{+46}:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

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 -8.6e+110)
   (/ (- b) a)
   (if (<= b -4.4e-210)
     (/ (- (sqrt (+ (* b b) (* (* a c) -4.0))) b) (* a 2.0))
     (if (<= b 5.161151587419122e+46)
       (/ (* 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 <= -8.6e+110) {
		tmp = -b / a;
	} else if (b <= -4.4e-210) {
		tmp = (sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
	} else if (b <= 5.161151587419122e+46) {
		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 <= -8.6e+110) {
		tmp = -b / a;
	} else if (b <= -4.4e-210) {
		tmp = (Math.sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
	} else if (b <= 5.161151587419122e+46) {
		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 <= -8.6e+110:
		tmp = -b / a
	elif b <= -4.4e-210:
		tmp = (math.sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0)
	elif b <= 5.161151587419122e+46:
		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(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.6e+110)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -4.4e-210)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0))) - b) / Float64(a * 2.0));
	elseif (b <= 5.161151587419122e+46)
		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 <= -8.6e+110)
		tmp = -b / a;
	elseif (b <= -4.4e-210)
		tmp = (sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
	elseif (b <= 5.161151587419122e+46)
		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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := If[LessEqual[b, -8.6e+110], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -4.4e-210], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.161151587419122e+46], 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])]]]

Target

Original	34.3
Target	21.1
Herbie	8.2

\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if b < -8.60000000000000014e110

   1. Initial program 50.8

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

   2. Simplified 50.8

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

   3. Applied egg-rr 35.9

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b\right) \cdot 0.5}}} \]

   4. Taylor expanded in b around -inf 3.6

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

   5. Simplified 3.6

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

   if -8.60000000000000014e110 < b < -4.39999999999999979e-210

   1. Initial program 6.4

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

   if -4.39999999999999979e-210 < b < 5.1611515874191217e46

   1. Initial program 26.7

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

   2. Simplified 26.8

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

   3. Applied egg-rr 27.5

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

   4. Taylor expanded in a around 0 15.5

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

   if 5.1611515874191217e46 < b

   1. Initial program 57.0

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

   2. Taylor expanded in b around inf 4.1

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

   3. Simplified 4.1

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.161151587419122 \cdot 10^{+46}:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.2
Cost	13964

\[\begin{array}{l} t_0 := \left(a \cdot c\right) \cdot -4\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.161151587419122 \cdot 10^{+46}:\\ \;\;\;\;\frac{-2}{\frac{b + \mathsf{hypot}\left(\sqrt{t_0}, b\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 2
Error	10.9
Cost	7888

\[\begin{array}{l} t_0 := \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 3
Error	10.8
Cost	7888

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 4
Error	14.3
Cost	7632

\[\begin{array}{l} t_0 := \frac{0.5}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\\ \mathbf{if}\;b \leq -1.22 \cdot 10^{-26}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 5
Error	14.3
Cost	7632

\[\begin{array}{l} t_0 := \frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.22 \cdot 10^{-26}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	22.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{-260}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 7
Error	40.3
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.3527077151374614 \cdot 10^{+70}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 8
Error	22.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-217}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 9
Error	62.3
Cost	192

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

Alternative 10
Error	56.7
Cost	192

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

Reproduce

herbie shell --seed 2022217 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))