The quadratic formula (r2)

Average Error: 34.2 → 10.4

Time: 24.7s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-42}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

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 -3.6e-42)
   (- (/ c b))
   (if (<= b 4.5e+50)
     (/ (+ 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 <= -3.6e-42) {
		tmp = -(c / b);
	} else if (b <= 4.5e+50) {
		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 <= (-3.6d-42)) then
        tmp = -(c / b)
    else if (b <= 4.5d+50) 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 <= -3.6e-42) {
		tmp = -(c / b);
	} else if (b <= 4.5e+50) {
		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 <= -3.6e-42:
		tmp = -(c / b)
	elif b <= 4.5e+50:
		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 <= -3.6e-42)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 4.5e+50)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(c * Float64(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 <= -3.6e-42)
		tmp = -(c / b);
	elseif (b <= 4.5e+50)
		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, -3.6e-42], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 4.5e+50], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]

Target

Original	34.2
Target	20.9
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 < -3.6000000000000002e-42

   1. Initial program 54.6

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

   2. Simplified 54.6

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

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

   3. Taylor expanded in b around -inf 7.2

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

   4. Simplified 7.2

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

      [Start] 7.2	\[ -1 \cdot \frac{c}{b} \]
      rational.json-simplify-2 [=>] 7.2	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational.json-simplify-9 [=>] 7.2	\[ \color{blue}{-\frac{c}{b}} \]

   if -3.6000000000000002e-42 < b < 4.50000000000000014e50

   1. Initial program 15.3

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

   2. Simplified 15.3

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

      [Start] 15.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 15.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 15.3

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{-\left(a + a\right)} + 0} \]

   4. Simplified 15.3

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

      [Start] 15.3	\[ \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{-\left(a + a\right)} + 0 \]

   if 4.50000000000000014e50 < b

   1. Initial program 37.6

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

   2. Simplified 37.6

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

      [Start] 37.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 37.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 5.9

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

   4. Simplified 5.9

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

      [Start] 5.9	\[ -1 \cdot \frac{b}{a} \]
      rational.json-simplify-2 [=>] 5.9	\[ \color{blue}{\frac{b}{a} \cdot -1} \]
      rational.json-simplify-9 [=>] 5.9	\[ \color{blue}{-\frac{b}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-42}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.5
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.08 \cdot 10^{-42}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+50}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 2
Error	10.4
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-41}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 3
Error	13.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-42}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-44}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	13.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-42}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-44}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{\sqrt{c \cdot \left(a \cdot -4\right)} + b}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	13.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-42}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} + b}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 6
Error	13.8
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{-42}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	13.8
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{-42}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 8
Error	19.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-162}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.96 \cdot 10^{-174}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 9
Error	39.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -0.07:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 10
Error	22.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-226}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 11
Error	56.5
Cost	192

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

Reproduce

herbie shell --seed 2023075 
(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)))