quadp (p42, positive)

Average Error: 33.8 → 10.7

Time: 20.0s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

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 -1.35e+154)
   (- (/ b a))
   (if (<= b 4.4e-148)
     (/ (+ (- b) (sqrt (- (* b b) (* a (* 4.0 c))))) (+ a 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 <= -1.35e+154) {
		tmp = -(b / a);
	} else if (b <= 4.4e-148) {
		tmp = (-b + sqrt(((b * b) - (a * (4.0 * c))))) / (a + 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 <= (-1.35d+154)) then
        tmp = -(b / a)
    else if (b <= 4.4d-148) then
        tmp = (-b + sqrt(((b * b) - (a * (4.0d0 * c))))) / (a + 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 <= -1.35e+154) {
		tmp = -(b / a);
	} else if (b <= 4.4e-148) {
		tmp = (-b + Math.sqrt(((b * b) - (a * (4.0 * c))))) / (a + 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 <= -1.35e+154:
		tmp = -(b / a)
	elif b <= 4.4e-148:
		tmp = (-b + math.sqrt(((b * b) - (a * (4.0 * c))))) / (a + 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 <= -1.35e+154)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 4.4e-148)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(a * Float64(4.0 * c))))) / Float64(a + a));
	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 <= -1.35e+154)
		tmp = -(b / a);
	elseif (b <= 4.4e-148)
		tmp = (-b + sqrt(((b * b) - (a * (4.0 * c))))) / (a + 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, -1.35e+154], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 4.4e-148], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(4.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Target

Original	33.8
Target	20.8
Herbie	10.7

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

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best_45_simplify-25 [=>] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{a \cdot \left(4 \cdot c\right)}}}{2 \cdot a} \]
      rational_best_45_simplify-91 [=>] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{a \cdot 2}} \]
      metadata-eval [<=] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \color{blue}{\left(1 + 1\right)}} \]
      metadata-eval [<=] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \left(\color{blue}{\frac{4}{4}} + 1\right)} \]
      metadata-eval [<=] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \left(\frac{4}{4} + \color{blue}{\frac{4}{4}}\right)} \]
      rational_best_45_simplify-71 [<=] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{\frac{4}{4} \cdot a + a \cdot \frac{4}{4}}} \]
      rational_best_45_simplify-91 [<=] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{a \cdot \frac{4}{4}} + a \cdot \frac{4}{4}} \]
      rational_best_45_simplify-91 [=>] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \frac{4}{4} + \color{blue}{\frac{4}{4} \cdot a}} \]
      rational_best_45_simplify-71 [=>] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{\frac{4}{4} \cdot \left(a + a\right)}} \]
      rational_best_45_simplify-91 [=>] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{\left(a + a\right) \cdot \frac{4}{4}}} \]
      metadata-eval [=>] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\left(a + a\right) \cdot \color{blue}{1}} \]
      rational_best_45_simplify-8 [=>] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{a + a}} \]

   3. Taylor expanded in b around -inf 1.9

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

   4. Simplified 1.9

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

      [Start] 1.9	\[ -1 \cdot \frac{b}{a} \]
      rational_best_45_simplify-91 [=>] 1.9	\[ \color{blue}{\frac{b}{a} \cdot -1} \]
      rational_best_45_simplify-16 [=>] 1.9	\[ \color{blue}{-\frac{b}{a}} \]

   if -1.35000000000000003e154 < b < 4.40000000000000034e-148

   1. Initial program 10.2

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

   2. Simplified 10.2

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

      [Start] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best_45_simplify-25 [=>] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{a \cdot \left(4 \cdot c\right)}}}{2 \cdot a} \]
      rational_best_45_simplify-91 [=>] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{a \cdot 2}} \]
      metadata-eval [<=] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \color{blue}{\left(1 + 1\right)}} \]
      metadata-eval [<=] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \left(\color{blue}{\frac{4}{4}} + 1\right)} \]
      metadata-eval [<=] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \left(\frac{4}{4} + \color{blue}{\frac{4}{4}}\right)} \]
      rational_best_45_simplify-71 [<=] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{\frac{4}{4} \cdot a + a \cdot \frac{4}{4}}} \]
      rational_best_45_simplify-91 [<=] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{a \cdot \frac{4}{4}} + a \cdot \frac{4}{4}} \]
      rational_best_45_simplify-91 [=>] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \frac{4}{4} + \color{blue}{\frac{4}{4} \cdot a}} \]
      rational_best_45_simplify-71 [=>] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{\frac{4}{4} \cdot \left(a + a\right)}} \]
      rational_best_45_simplify-91 [=>] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{\left(a + a\right) \cdot \frac{4}{4}}} \]
      metadata-eval [=>] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\left(a + a\right) \cdot \color{blue}{1}} \]
      rational_best_45_simplify-8 [=>] 10.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{a + a}} \]

   if 4.40000000000000034e-148 < b

   1. Initial program 49.1

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

   2. Simplified 49.1

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

      [Start] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best_45_simplify-25 [=>] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{a \cdot \left(4 \cdot c\right)}}}{2 \cdot a} \]
      rational_best_45_simplify-91 [=>] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{a \cdot 2}} \]
      metadata-eval [<=] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \color{blue}{\left(1 + 1\right)}} \]
      metadata-eval [<=] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \left(\color{blue}{\frac{4}{4}} + 1\right)} \]
      metadata-eval [<=] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \left(\frac{4}{4} + \color{blue}{\frac{4}{4}}\right)} \]
      rational_best_45_simplify-71 [<=] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{\frac{4}{4} \cdot a + a \cdot \frac{4}{4}}} \]
      rational_best_45_simplify-91 [<=] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{a \cdot \frac{4}{4}} + a \cdot \frac{4}{4}} \]
      rational_best_45_simplify-91 [=>] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a \cdot \frac{4}{4} + \color{blue}{\frac{4}{4} \cdot a}} \]
      rational_best_45_simplify-71 [=>] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{\frac{4}{4} \cdot \left(a + a\right)}} \]
      rational_best_45_simplify-91 [=>] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{\left(a + a\right) \cdot \frac{4}{4}}} \]
      metadata-eval [=>] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\left(a + a\right) \cdot \color{blue}{1}} \]
      rational_best_45_simplify-8 [=>] 49.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{\color{blue}{a + a}} \]

   3. Taylor expanded in b around inf 13.4

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

   4. Simplified 13.4

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

      [Start] 13.4	\[ -1 \cdot \frac{c}{b} \]
      rational_best_45_simplify-91 [=>] 13.4	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational_best_45_simplify-16 [=>] 13.4	\[ \color{blue}{-\frac{c}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.8
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+148}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 2
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-91}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 3
Error	23.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 4
Error	39.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 5
Error	23.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-260}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	56.4
Cost	192

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

Reproduce

herbie shell --seed 2023098 
(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)))