quadm (p42, negative)

Average Error: 53.48% → 15.56%

Time: 18.6s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -4.2e-62)
   (/ (- c) b)
   (if (<= b 1.1e+125)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* 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 <= -4.2e-62) {
		tmp = -c / b;
	} else if (b <= 1.1e+125) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (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 <= (-4.2d-62)) then
        tmp = -c / b
    else if (b <= 1.1d+125) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (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 <= -4.2e-62) {
		tmp = -c / b;
	} else if (b <= 1.1e+125) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (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 <= -4.2e-62:
		tmp = -c / b
	elif b <= 1.1e+125:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (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 <= -4.2e-62)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.1e+125)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / 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 <= -4.2e-62)
		tmp = -c / b;
	elseif (b <= 1.1e+125)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (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, -4.2e-62], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.1e+125], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	53.48%
Target	32.79%
Herbie	15.56%

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

Derivation

1. Split input into 3 regimes

2. if b < -4.1999999999999998e-62

   1. Initial program 84.38

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

   2. Simplified 84.38

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, \frac{a \cdot c}{-0.25}\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof

      [Start] 84.38	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 84.38	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 84.38	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 84.38	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 84.38	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 84.38	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 84.38	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 84.38	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around -inf 13.8

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

   4. Simplified 13.8

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

      [Start] 13.8	\[ -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 13.8	\[ \color{blue}{-\frac{c}{b}} \]
      distribute-neg-frac [=>] 13.8	\[ \color{blue}{\frac{-c}{b}} \]

   if -4.1999999999999998e-62 < b < 1.09999999999999995e125

   1. Initial program 20.09

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

   if 1.09999999999999995e125 < b

   1. Initial program 85.27

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

   2. Simplified 85.34

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, \frac{a \cdot c}{-0.25}\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof

      [Start] 85.27	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 85.27	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 85.27	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 85.27	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 85.34	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 85.34	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 85.34	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 85.34	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around inf 16.28

      \[\leadsto \left(b + \color{blue}{\left(b + -2 \cdot \frac{c \cdot a}{b}\right)}\right) \cdot \frac{-0.5}{a} \]

   4. Simplified 16.28

      \[\leadsto \left(b + \color{blue}{\left(b + \frac{\left(c \cdot a\right) \cdot -2}{b}\right)}\right) \cdot \frac{-0.5}{a} \]
      Proof

      [Start] 16.28	\[ \left(b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right) \cdot \frac{-0.5}{a} \]
      associate-*r/ [=>] 16.28	\[ \left(b + \left(b + \color{blue}{\frac{-2 \cdot \left(c \cdot a\right)}{b}}\right)\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 16.28	\[ \left(b + \left(b + \frac{\color{blue}{\left(c \cdot a\right) \cdot -2}}{b}\right)\right) \cdot \frac{-0.5}{a} \]

   5. Taylor expanded in b around 0 3.83

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

   6. Simplified 3.83

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

      [Start] 3.83	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 3.83	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 3.83	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 15.56

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.71%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+108}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	21.02%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.05 \cdot 10^{-117}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	20.98%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.05 \cdot 10^{-117}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	35.71%
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 5
Error	62.58%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 6
Error	35.81%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 7
Error	88.59%
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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)))