The quadratic formula (r2)

Average Error: 33.7 → 10.4

Time: 22.6s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{-54}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

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 -3.9e-54)
   (- (/ c b))
   (if (<= b 8e+36)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a))
     (+ (- (/ b 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 <= -3.9e-54) {
		tmp = -(c / b);
	} else if (b <= 8e+36) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} else {
		tmp = -(b / a) + (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 <= (-3.9d-54)) then
        tmp = -(c / b)
    else if (b <= 8d+36) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
    else
        tmp = -(b / a) + (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 <= -3.9e-54) {
		tmp = -(c / b);
	} else if (b <= 8e+36) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} else {
		tmp = -(b / a) + (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 <= -3.9e-54:
		tmp = -(c / b)
	elif b <= 8e+36:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)
	else:
		tmp = -(b / a) + (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 <= -3.9e-54)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 8e+36)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-Float64(b / a)) + 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 <= -3.9e-54)
		tmp = -(c / b);
	elseif (b <= 8e+36)
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	else
		tmp = -(b / a) + (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, -3.9e-54], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 8e+36], 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], N[((-N[(b / a), $MachinePrecision]) + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Target

Original	33.7
Target	20.6
Herbie	10.4

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

Derivation

1. Split input into 3 regimes

2. if b < -3.9e-54

   1. Initial program 54.0

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

   2. Simplified 54.0

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

      [Start] 54.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 54.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 8.1

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

   4. Simplified 8.1

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

      [Start] 8.1	\[ -1 \cdot \frac{c}{b} \]
      rational.json-simplify-2 [=>] 8.1	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational.json-simplify-9 [=>] 8.1	\[ \color{blue}{-\frac{c}{b}} \]

   if -3.9e-54 < b < 8.00000000000000034e36

   1. Initial program 14.5

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

   if 8.00000000000000034e36 < b

   1. Initial program 34.8

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

   2. Simplified 34.8

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

      [Start] 34.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 34.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 6.1

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

   4. Simplified 6.1

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

      [Start] 6.1	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational.json-simplify-1 [=>] 6.1	\[ \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
      rational.json-simplify-2 [=>] 6.1	\[ \color{blue}{\frac{b}{a} \cdot -1} + \frac{c}{b} \]
      rational.json-simplify-9 [=>] 6.1	\[ \color{blue}{\left(-\frac{b}{a}\right)} + \frac{c}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{-54}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.3
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-49}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 2
Error	13.8
Cost	7304

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-53}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 3
Error	20.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.36 \cdot 10^{-124}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 4
Error	22.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -3.35 \cdot 10^{-264}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 5
Error	45.5
Cost	256

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

Reproduce

herbie shell --seed 2023077 
(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)))