The quadratic formula (r2)

Average Error: 34.0 → 10.0

Time: 17.7s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{-c}{b - \frac{c}{b} \cdot a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 -4.3e-76)
   (/ (- c) (- b (* (/ c b) a)))
   (if (<= b 1.6e+101)
     (/ (- (- b) (sqrt (+ (* b b) (* (* c a) -4.0)))) (* a 2.0))
     (/ (- b) a))))

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 <= -4.3e-76) {
		tmp = -c / (b - ((c / b) * a));
	} else if (b <= 1.6e+101) {
		tmp = (-b - sqrt(((b * b) + ((c * a) * -4.0)))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	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 <= (-4.3d-76)) then
        tmp = -c / (b - ((c / b) * a))
    else if (b <= 1.6d+101) then
        tmp = (-b - sqrt(((b * b) + ((c * a) * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = -b / a
    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 <= -4.3e-76) {
		tmp = -c / (b - ((c / b) * a));
	} else if (b <= 1.6e+101) {
		tmp = (-b - Math.sqrt(((b * b) + ((c * a) * -4.0)))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	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 <= -4.3e-76:
		tmp = -c / (b - ((c / b) * a))
	elif b <= 1.6e+101:
		tmp = (-b - math.sqrt(((b * b) + ((c * a) * -4.0)))) / (a * 2.0)
	else:
		tmp = -b / a
	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 <= -4.3e-76)
		tmp = Float64(Float64(-c) / Float64(b - Float64(Float64(c / b) * a)));
	elseif (b <= 1.6e+101)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + Float64(Float64(c * a) * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	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 <= -4.3e-76)
		tmp = -c / (b - ((c / b) * a));
	elseif (b <= 1.6e+101)
		tmp = (-b - sqrt(((b * b) + ((c * a) * -4.0)))) / (a * 2.0);
	else
		tmp = -b / a;
	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, -4.3e-76], N[((-c) / N[(b - N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+101], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Target

Original	34.0
Target	20.8
Herbie	10.0

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

Derivation

1. Split input into 3 regimes

2. if b < -4.2999999999999999e-76

   1. Initial program 53.0

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

   2. Simplified 53.0

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

      [Start] 53.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 53.0	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 53.0	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 53.0	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 53.0	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 53.0	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 53.0	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 53.0	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 53.0	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 53.0	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 53.0	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 53.0	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 53.0	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 53.0	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 53.0	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Applied egg-rr 53.0

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

   4. Taylor expanded in b around -inf 19.9

      \[\leadsto \frac{\frac{-0.5}{a}}{\color{blue}{0.5 \cdot \frac{b}{c \cdot a} - 0.5 \cdot \frac{1}{b}}} \]

   5. Applied egg-rr 24.5

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

   6. Simplified 25.9

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

      [Start] 24.5	\[ \frac{\frac{-0.5}{0.5 \cdot a}}{b - \frac{c}{b} \cdot a} \cdot \left(\left(\frac{c}{b} \cdot a\right) \cdot b\right) \]
      associate-*l/ [=>] 19.8	\[ \color{blue}{\frac{\frac{-0.5}{0.5 \cdot a} \cdot \left(\left(\frac{c}{b} \cdot a\right) \cdot b\right)}{b - \frac{c}{b} \cdot a}} \]
      associate-/r* [=>] 19.8	\[ \frac{\color{blue}{\frac{\frac{-0.5}{0.5}}{a}} \cdot \left(\left(\frac{c}{b} \cdot a\right) \cdot b\right)}{b - \frac{c}{b} \cdot a} \]
      metadata-eval [=>] 19.8	\[ \frac{\frac{\color{blue}{-1}}{a} \cdot \left(\left(\frac{c}{b} \cdot a\right) \cdot b\right)}{b - \frac{c}{b} \cdot a} \]
      associate-*l/ [=>] 19.8	\[ \frac{\color{blue}{\frac{-1 \cdot \left(\left(\frac{c}{b} \cdot a\right) \cdot b\right)}{a}}}{b - \frac{c}{b} \cdot a} \]
      *-commutative [=>] 19.8	\[ \frac{\frac{-1 \cdot \color{blue}{\left(b \cdot \left(\frac{c}{b} \cdot a\right)\right)}}{a}}{b - \frac{c}{b} \cdot a} \]
      associate-*l* [<=] 19.8	\[ \frac{\frac{\color{blue}{\left(-1 \cdot b\right) \cdot \left(\frac{c}{b} \cdot a\right)}}{a}}{b - \frac{c}{b} \cdot a} \]
      neg-mul-1 [<=] 19.8	\[ \frac{\frac{\color{blue}{\left(-b\right)} \cdot \left(\frac{c}{b} \cdot a\right)}{a}}{b - \frac{c}{b} \cdot a} \]
      *-commutative [<=] 19.8	\[ \frac{\frac{\color{blue}{\left(\frac{c}{b} \cdot a\right) \cdot \left(-b\right)}}{a}}{b - \frac{c}{b} \cdot a} \]
      associate-*l* [=>] 25.9	\[ \frac{\frac{\color{blue}{\frac{c}{b} \cdot \left(a \cdot \left(-b\right)\right)}}{a}}{b - \frac{c}{b} \cdot a} \]

   7. Taylor expanded in c around 0 9.0

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

   8. Simplified 9.0

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

      [Start] 9.0	\[ \frac{-1 \cdot c}{b - \frac{c}{b} \cdot a} \]
      mul-1-neg [=>] 9.0	\[ \frac{\color{blue}{-c}}{b - \frac{c}{b} \cdot a} \]

   if -4.2999999999999999e-76 < b < 1.60000000000000003e101

   1. Initial program 12.9

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

   if 1.60000000000000003e101 < b

   1. Initial program 46.6

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

   2. Simplified 46.6

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

      [Start] 46.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 46.6	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 46.6	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 46.6	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 46.6	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 46.6	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 46.6	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 46.6	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 46.6	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 46.6	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 46.6	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 46.6	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 46.6	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 46.6	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 46.6	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Taylor expanded in a around 0 4.4

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

   4. Simplified 4.4

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

      [Start] 4.4	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 4.4	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 4.4	\[ \frac{\color{blue}{-b}}{a} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{-c}{b - \frac{c}{b} \cdot a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.1
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{-c}{b - \frac{c}{b} \cdot a}\\ \mathbf{elif}\;b \leq 270:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	14.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-c}{b - \frac{c}{b} \cdot a}\\ \mathbf{elif}\;b \leq 245:\\ \;\;\;\;\left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	22.1
Cost	772

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b - \frac{c}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4
Error	39.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{-63}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Error	22.3
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	62.3
Cost	192

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

Alternative 7
Error	56.3
Cost	192

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

Reproduce

herbie shell --seed 2023025 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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