quadp (p42, positive)

Average Error: 34.5 → 9.9

Time: 12.5s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{-a} + \left(a + a \cdot -3\right) \cdot \left(-0.5 \cdot \frac{\frac{c}{b}}{a}\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

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 -3.5e+152)
   (+ (/ b (- a)) (* (+ a (* a -3.0)) (* -0.5 (/ (/ c b) a))))
   (if (<= b 1.1e-60)
     (/ (+ (- 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 <= -3.5e+152) {
		tmp = (b / -a) + ((a + (a * -3.0)) * (-0.5 * ((c / b) / a)));
	} else if (b <= 1.1e-60) {
		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 <= (-3.5d+152)) then
        tmp = (b / -a) + ((a + (a * (-3.0d0))) * ((-0.5d0) * ((c / b) / a)))
    else if (b <= 1.1d-60) 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 <= -3.5e+152) {
		tmp = (b / -a) + ((a + (a * -3.0)) * (-0.5 * ((c / b) / a)));
	} else if (b <= 1.1e-60) {
		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 <= -3.5e+152:
		tmp = (b / -a) + ((a + (a * -3.0)) * (-0.5 * ((c / b) / a)))
	elif b <= 1.1e-60:
		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 <= -3.5e+152)
		tmp = Float64(Float64(b / Float64(-a)) + Float64(Float64(a + Float64(a * -3.0)) * Float64(-0.5 * Float64(Float64(c / b) / a))));
	elseif (b <= 1.1e-60)
		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 <= -3.5e+152)
		tmp = (b / -a) + ((a + (a * -3.0)) * (-0.5 * ((c / b) / a)));
	elseif (b <= 1.1e-60)
		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, -3.5e+152], N[(N[(b / (-a)), $MachinePrecision] + N[(N[(a + N[(a * -3.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(c / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e-60], 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	34.5
Target	21.4
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 < -3.49999999999999981e152

   1. Initial program 63.2

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

   2. Simplified 63.2

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

      [Start] 63.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best-simplify-2 [=>] 63.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 10.2

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{a \cdot 2} \]

   4. Simplified 2.5

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

      [Start] 10.2	\[ \frac{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}{a \cdot 2} \]
      rational_best-simplify-2 [<=] 10.2	\[ \frac{2 \cdot \frac{\color{blue}{a \cdot c}}{b} + -2 \cdot b}{a \cdot 2} \]
      rational_best-simplify-49 [=>] 2.5	\[ \frac{2 \cdot \color{blue}{\left(c \cdot \frac{a}{b}\right)} + -2 \cdot b}{a \cdot 2} \]
      rational_best-simplify-46 [=>] 2.5	\[ \frac{\color{blue}{c \cdot \left(2 \cdot \frac{a}{b}\right)} + -2 \cdot b}{a \cdot 2} \]
      rational_best-simplify-2 [<=] 2.5	\[ \frac{c \cdot \left(2 \cdot \frac{a}{b}\right) + \color{blue}{b \cdot -2}}{a \cdot 2} \]

   5. Applied egg-rr 2.5

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

   6. Taylor expanded in b around -inf 10.2

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c \cdot \left(a + \left(-2 \cdot a + -1 \cdot a\right)\right)}{a \cdot b} + -1 \cdot \frac{b}{a}} \]

   7. Simplified 2.5

      \[\leadsto \color{blue}{\frac{b}{-a} + \left(a + a \cdot -3\right) \cdot \left(-0.5 \cdot \frac{\frac{c}{b}}{a}\right)} \]
      Proof

      [Start] 10.2	\[ -0.5 \cdot \frac{c \cdot \left(a + \left(-2 \cdot a + -1 \cdot a\right)\right)}{a \cdot b} + -1 \cdot \frac{b}{a} \]
      rational_best-simplify-1 [=>] 10.2	\[ \color{blue}{-1 \cdot \frac{b}{a} + -0.5 \cdot \frac{c \cdot \left(a + \left(-2 \cdot a + -1 \cdot a\right)\right)}{a \cdot b}} \]
      rational_best-simplify-10 [=>] 10.2	\[ \color{blue}{\left(-\frac{b}{a}\right)} + -0.5 \cdot \frac{c \cdot \left(a + \left(-2 \cdot a + -1 \cdot a\right)\right)}{a \cdot b} \]
      rational_best-simplify-9 [=>] 10.2	\[ \color{blue}{\frac{\frac{b}{a}}{-1}} + -0.5 \cdot \frac{c \cdot \left(a + \left(-2 \cdot a + -1 \cdot a\right)\right)}{a \cdot b} \]
      rational_best-simplify-48 [=>] 10.2	\[ \color{blue}{\frac{b}{-1 \cdot a}} + -0.5 \cdot \frac{c \cdot \left(a + \left(-2 \cdot a + -1 \cdot a\right)\right)}{a \cdot b} \]
      rational_best-simplify-10 [=>] 10.2	\[ \frac{b}{\color{blue}{-a}} + -0.5 \cdot \frac{c \cdot \left(a + \left(-2 \cdot a + -1 \cdot a\right)\right)}{a \cdot b} \]
      rational_best-simplify-49 [=>] 2.6	\[ \frac{b}{-a} + -0.5 \cdot \color{blue}{\left(\left(a + \left(-2 \cdot a + -1 \cdot a\right)\right) \cdot \frac{c}{a \cdot b}\right)} \]
      rational_best-simplify-46 [=>] 2.6	\[ \frac{b}{-a} + \color{blue}{\left(a + \left(-2 \cdot a + -1 \cdot a\right)\right) \cdot \left(-0.5 \cdot \frac{c}{a \cdot b}\right)} \]
      rational_best-simplify-2 [=>] 2.6	\[ \frac{b}{-a} + \left(a + \left(\color{blue}{a \cdot -2} + -1 \cdot a\right)\right) \cdot \left(-0.5 \cdot \frac{c}{a \cdot b}\right) \]
      rational_best-simplify-2 [=>] 2.6	\[ \frac{b}{-a} + \left(a + \left(a \cdot -2 + \color{blue}{a \cdot -1}\right)\right) \cdot \left(-0.5 \cdot \frac{c}{a \cdot b}\right) \]
      rational_best-simplify-53 [=>] 2.6	\[ \frac{b}{-a} + \left(a + \color{blue}{a \cdot \left(-1 + -2\right)}\right) \cdot \left(-0.5 \cdot \frac{c}{a \cdot b}\right) \]
      metadata-eval [=>] 2.6	\[ \frac{b}{-a} + \left(a + a \cdot \color{blue}{-3}\right) \cdot \left(-0.5 \cdot \frac{c}{a \cdot b}\right) \]
      rational_best-simplify-48 [<=] 2.5	\[ \frac{b}{-a} + \left(a + a \cdot -3\right) \cdot \left(-0.5 \cdot \color{blue}{\frac{\frac{c}{b}}{a}}\right) \]

   if -3.49999999999999981e152 < b < 1.0999999999999999e-60

   1. Initial program 12.9

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

   if 1.0999999999999999e-60 < b

   1. Initial program 54.1

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

   2. Simplified 54.1

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

      [Start] 54.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best-simplify-2 [=>] 54.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 8.1

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

   4. Simplified 8.1

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

      [Start] 8.1	\[ -1 \cdot \frac{c}{b} \]
      rational_best-simplify-10 [=>] 8.1	\[ \color{blue}{-\frac{c}{b}} \]
      rational_best-simplify-9 [=>] 8.1	\[ \color{blue}{\frac{\frac{c}{b}}{-1}} \]
      rational_best-simplify-48 [=>] 8.1	\[ \color{blue}{\frac{c}{-1 \cdot b}} \]
      rational_best-simplify-10 [=>] 8.1	\[ \frac{c}{\color{blue}{-b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{-a} + \left(a + a \cdot -3\right) \cdot \left(-0.5 \cdot \frac{\frac{c}{b}}{a}\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.3
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} + \left(-b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 2
Error	19.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{-108}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 3
Error	22.7
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-241}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 4
Error	39.4
Cost	256

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

Alternative 5
Error	56.4
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023097 
(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)))