The quadratic formula (r1)

Average Error: 33.6 → 10.5

Time: 6.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -3.5e+152)
   (+ (/ c b) (/ (- b) a))
   (if (<= b 6.5e-18)
     (/ (+ (- 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 = (c / b) + (-b / a);
	} else if (b <= 6.5e-18) {
		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 = (c / b) + (-b / a)
    else if (b <= 6.5d-18) 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 = (c / b) + (-b / a);
	} else if (b <= 6.5e-18) {
		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 = (c / b) + (-b / a)
	elif b <= 6.5e-18:
		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 <= -3.5e+152)
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	elseif (b <= 6.5e-18)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * 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 <= -3.5e+152)
		tmp = (c / b) + (-b / a);
	elseif (b <= 6.5e-18)
		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, -3.5e+152], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-18], 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	33.6
Target	20.4
Herbie	10.5

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

Derivation

1. Split input into 3 regimes

2. if b < -3.49999999999999981e152

   1. Initial program 63.4

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

   2. Taylor expanded in b around -inf 2.1

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

   3. Simplified 2.1

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

   if -3.49999999999999981e152 < b < 6.50000000000000008e-18

   1. Initial program 14.5

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

   if 6.50000000000000008e-18 < b

   1. Initial program 54.9

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

   2. Taylor expanded in b around inf 6.5

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

   3. Simplified 6.5

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022203 
(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)))