quadp (p42, positive)

Average Error: 33.5 → 9.9

Time: 16.0s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} + \frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

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 -1.2e+151)
   (+ (/ c b) (/ b (- a)))
   (if (<= b 1.35e-43)
     (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a))
     (/ 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 <= -1.2e+151) {
		tmp = (c / b) + (b / -a);
	} else if (b <= 1.35e-43) {
		tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.2d+151)) then
        tmp = (c / b) + (b / -a)
    else if (b <= 1.35d-43) then
        tmp = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

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 <= -1.2e+151) {
		tmp = (c / b) + (b / -a);
	} else if (b <= 1.35e-43) {
		tmp = (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} 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 <= -1.2e+151:
		tmp = (c / b) + (b / -a)
	elif b <= 1.35e-43:
		tmp = (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)
	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 <= -1.2e+151)
		tmp = Float64(Float64(c / b) + Float64(b / Float64(-a)));
	elseif (b <= 1.35e-43)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(-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 <= -1.2e+151)
		tmp = (c / b) + (b / -a);
	elseif (b <= 1.35e-43)
		tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	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, -1.2e+151], N[(N[(c / b), $MachinePrecision] + N[(b / (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-43], 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], N[(c / (-b)), $MachinePrecision]]]

Target

Original	33.5
Target	20.4
Herbie	9.9

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

Derivation

1. Split input into 3 regimes

2. if b < -1.20000000000000005e151

   1. Initial program 63.3

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

   2. Simplified 63.3

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

      [Start] 63.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 63.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 2.2

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

   4. Simplified 2.2

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

      [Start] 2.2	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational.json-simplify-2 [=>] 2.2	\[ \frac{c}{b} + \color{blue}{\frac{b}{a} \cdot -1} \]
      rational.json-simplify-9 [=>] 2.2	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      rational.json-simplify-10 [=>] 2.2	\[ \frac{c}{b} + \color{blue}{\frac{\frac{b}{a}}{-1}} \]
      rational.json-simplify-47 [=>] 2.2	\[ \frac{c}{b} + \color{blue}{\frac{b}{a \cdot -1}} \]
      rational.json-simplify-8 [<=] 2.2	\[ \frac{c}{b} + \frac{b}{\color{blue}{-a}} \]

   if -1.20000000000000005e151 < b < 1.34999999999999996e-43

   1. Initial program 13.1

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

   if 1.34999999999999996e-43 < b

   1. Initial program 53.6

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

   2. Simplified 53.6

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

      [Start] 53.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 53.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 7.8

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

   4. Simplified 7.8

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

      [Start] 7.8	\[ -1 \cdot \frac{c}{b} \]
      rational.json-simplify-2 [=>] 7.8	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational.json-simplify-9 [=>] 7.8	\[ \color{blue}{-\frac{c}{b}} \]
      rational.json-simplify-10 [=>] 7.8	\[ \color{blue}{\frac{\frac{c}{b}}{-1}} \]
      rational.json-simplify-47 [=>] 7.8	\[ \color{blue}{\frac{c}{b \cdot -1}} \]
      rational.json-simplify-8 [<=] 7.8	\[ \frac{c}{\color{blue}{-b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} + \frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+150}:\\ \;\;\;\;\frac{c}{b} + \frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 2
Error	10.0
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{c}{b} + \frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} + \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 3
Error	13.6
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{c}{b} + \frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-43}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} + \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 4
Error	20.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{c}{b} + \frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 5
Error	39.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	23.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-230}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 7
Error	56.6
Cost	192

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

Reproduce

herbie shell --seed 2023073 
(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)))