The quadratic formula (r1)

Average Error: 33.9 → 10.4

Time: 20.0s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+73}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

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 -1.15e+73)
   (/ (- b) a)
   (if (<= b 1.95e-25)
     (/ (+ (- 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 <= -1.15e+73) {
		tmp = -b / a;
	} else if (b <= 1.95e-25) {
		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 <= (-1.15d+73)) then
        tmp = -b / a
    else if (b <= 1.95d-25) 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 <= -1.15e+73) {
		tmp = -b / a;
	} else if (b <= 1.95e-25) {
		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 <= -1.15e+73:
		tmp = -b / a
	elif b <= 1.95e-25:
		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(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e+73)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.95e-25)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * 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 <= -1.15e+73)
		tmp = -b / a;
	elseif (b <= 1.95e-25)
		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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := If[LessEqual[b, -1.15e+73], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.95e-25], 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], N[(c / (-b)), $MachinePrecision]]]

Target

Original	33.9
Target	20.8
Herbie	10.4

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

Derivation

1. Split input into 3 regimes

2. if b < -1.15e73

   1. Initial program 42.2

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

   2. Simplified 42.2

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

      [Start] 42.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational_best-simplify-2 [=>] 42.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 4.4

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

   4. Simplified 4.4

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

      [Start] 4.4	\[ -1 \cdot \frac{b}{a} \]
      rational_best-simplify-2 [=>] 4.4	\[ \color{blue}{\frac{b}{a} \cdot -1} \]
      rational_best-simplify-12 [=>] 4.4	\[ \color{blue}{-\frac{b}{a}} \]
      rational_best-simplify-9 [=>] 4.4	\[ \color{blue}{\frac{\frac{b}{a}}{-1}} \]
      rational_best-simplify-45 [=>] 4.4	\[ \color{blue}{\frac{\frac{b}{-1}}{a}} \]
      rational_best-simplify-9 [<=] 4.4	\[ \frac{\color{blue}{-b}}{a} \]

   if -1.15e73 < b < 1.95e-25

   1. Initial program 15.4

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

   if 1.95e-25 < b

   1. Initial program 54.6

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

   2. Simplified 54.6

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

      [Start] 54.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational_best-simplify-2 [=>] 54.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 6.7

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

   4. Simplified 6.7

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

      [Start] 6.7	\[ -1 \cdot \frac{c}{b} \]
      rational_best-simplify-2 [=>] 6.7	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational_best-simplify-12 [=>] 6.7	\[ \color{blue}{-\frac{c}{b}} \]
      rational_best-simplify-9 [=>] 6.7	\[ \color{blue}{\frac{\frac{c}{b}}{-1}} \]
      rational_best-simplify-48 [=>] 6.7	\[ \color{blue}{\frac{c}{-1 \cdot b}} \]
      rational_best-simplify-2 [=>] 6.7	\[ \frac{c}{\color{blue}{b \cdot -1}} \]
      rational_best-simplify-13 [<=] 6.7	\[ \frac{c}{\color{blue}{-b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+73}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.1
Cost	7560

\[\begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{c}{-b}\\ \mathbf{if}\;b \leq -5 \cdot 10^{-84}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{a + a} \cdot \left(t_0 + \left(-b\right)\right)\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.1
Cost	7504

\[\begin{array}{l} t_0 := \frac{c}{-b}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} + \left(-b\right)}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	14.5
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 4
Error	20.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-180}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 5
Error	22.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.76 \cdot 10^{-217}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]

Alternative 6
Error	39.9
Cost	256

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

Alternative 7
Error	56.5
Cost	192

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

Reproduce

herbie shell --seed 2023096 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))