quadm (p42, negative)

Average Error: 34.6 → 10.0

Time: 27.8s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-113}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

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 -9e-113)
   (- (/ c b))
   (if (<= b 9.5e+125)
     (/ (+ 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 <= -9e-113) {
		tmp = -(c / b);
	} else if (b <= 9.5e+125) {
		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 <= (-9d-113)) then
        tmp = -(c / b)
    else if (b <= 9.5d+125) 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 <= -9e-113) {
		tmp = -(c / b);
	} else if (b <= 9.5e+125) {
		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 <= -9e-113:
		tmp = -(c / b)
	elif b <= 9.5e+125:
		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 <= -9e-113)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 9.5e+125)
		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 <= -9e-113)
		tmp = -(c / b);
	elseif (b <= 9.5e+125)
		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, -9e-113], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 9.5e+125], 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	34.6
Target	21.0
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 < -9.0000000000000002e-113

   1. Initial program 51.6

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

   2. Simplified 51.6

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

      [Start] 51.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 51.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 10.7

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

   4. Simplified 10.7

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

      [Start] 10.7	\[ -1 \cdot \frac{c}{b} \]
      rational.json-simplify-2 [=>] 10.7	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational.json-simplify-9 [=>] 10.7	\[ \color{blue}{-\frac{c}{b}} \]

   if -9.0000000000000002e-113 < b < 9.50000000000000041e125

   1. Initial program 11.3

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

   2. Simplified 11.3

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

      [Start] 11.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 11.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 11.3

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

   4. Simplified 11.3

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

      [Start] 11.3	\[ \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{-\left(a + a\right)} + 0 \]

   if 9.50000000000000041e125 < b

   1. Initial program 54.3

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

   2. Simplified 54.3

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

      [Start] 54.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 54.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 3.4

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

   4. Simplified 3.4

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

      [Start] 3.4	\[ -1 \cdot \frac{b}{a} \]
      rational.json-simplify-2 [=>] 3.4	\[ \color{blue}{\frac{b}{a} \cdot -1} \]
      rational.json-simplify-9 [=>] 3.4	\[ \color{blue}{-\frac{b}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-113}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-113}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 2
Error	13.5
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-113}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 3
Error	13.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-113}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-99}:\\ \;\;\;\;\left(\sqrt{c \cdot \left(a \cdot -4\right)} + b\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	13.7
Cost	7304

\[\begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-113}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-104}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	19.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-152}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-104}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternative 6
Error	22.5
Cost	644

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

Alternative 7
Error	40.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1050000000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 8
Error	22.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-221}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 9
Error	56.7
Cost	192

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

Reproduce

herbie shell --seed 2023068 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))