The quadratic formula (r2)

Average Accuracy: 46.9% → 84.4%

Time: 18.2s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{-62}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -2.35e-62)
   (/ (- c) b)
   (if (<= b 9e+102)
     (/ (- (- b) (sqrt (+ (* b b) (* (* c a) -4.0)))) (* a 2.0))
     (- (/ c b) (/ 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 <= -2.35e-62) {
		tmp = -c / b;
	} else if (b <= 9e+102) {
		tmp = (-b - sqrt(((b * b) + ((c * a) * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (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 <= (-2.35d-62)) then
        tmp = -c / b
    else if (b <= 9d+102) then
        tmp = (-b - sqrt(((b * b) + ((c * a) * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = (c / b) - (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 <= -2.35e-62) {
		tmp = -c / b;
	} else if (b <= 9e+102) {
		tmp = (-b - Math.sqrt(((b * b) + ((c * a) * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (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 <= -2.35e-62:
		tmp = -c / b
	elif b <= 9e+102:
		tmp = (-b - math.sqrt(((b * b) + ((c * a) * -4.0)))) / (a * 2.0)
	else:
		tmp = (c / b) - (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 <= -2.35e-62)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 9e+102)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + Float64(Float64(c * a) * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - 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 <= -2.35e-62)
		tmp = -c / b;
	elseif (b <= 9e+102)
		tmp = (-b - sqrt(((b * b) + ((c * a) * -4.0)))) / (a * 2.0);
	else
		tmp = (c / b) - (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, -2.35e-62], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 9e+102], 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[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	46.9%
Target	66.7%
Herbie	84.4%

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

Derivation

1. Split input into 3 regimes

2. if b < -2.34999999999999992e-62

   1. Initial program 16.6%

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

   2. Taylor expanded in b around -inf 87.6%

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

   3. Simplified 87.6%

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

      [Start] 87.6	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 87.6	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 87.6	\[ \frac{\color{blue}{-c}}{b} \]

   if -2.34999999999999992e-62 < b < 9.00000000000000042e102

   1. Initial program 78.7%

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

   if 9.00000000000000042e102 < b

   1. Initial program 24.9%

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

   2. Taylor expanded in b around inf 94.0%

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

   3. Simplified 94.0%

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

      [Start] 94.0	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 94.0	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 94.0	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{-62}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.4%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+100}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Accuracy	79.2%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-59}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-62}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Accuracy	64.7%
Cost	580

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

Alternative 4
Accuracy	37.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Accuracy	64.7%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-225}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Accuracy	2.6%
Cost	192

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

Alternative 7
Accuracy	11.2%
Cost	192

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

Reproduce

herbie shell --seed 2023136 
(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)))