quadm (p42, negative)

Average Error: 34.1 → 10.7

Time: 16.5s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 1.62 \cdot 10^{+142}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 -6.4e-9)
   (/ 1.0 (- (/ a b) (/ b c)))
   (if (<= b 1.62e+142)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 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 <= -6.4e-9) {
		tmp = 1.0 / ((a / b) - (b / c));
	} else if (b <= 1.62e+142) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (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 <= (-6.4d-9)) then
        tmp = 1.0d0 / ((a / b) - (b / c))
    else if (b <= 1.62d+142) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (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 <= -6.4e-9) {
		tmp = 1.0 / ((a / b) - (b / c));
	} else if (b <= 1.62e+142) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (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 <= -6.4e-9:
		tmp = 1.0 / ((a / b) - (b / c))
	elif b <= 1.62e+142:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (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 <= -6.4e-9)
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	elseif (b <= 1.62e+142)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / 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 <= -6.4e-9)
		tmp = 1.0 / ((a / b) - (b / c));
	elseif (b <= 1.62e+142)
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (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, -6.4e-9], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.62e+142], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Target

Original	34.1
Target	20.8
Herbie	10.7

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

Derivation

1. Split input into 3 regimes

2. if b < -6.40000000000000023e-9

   1. Initial program 55.0

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

   2. Simplified 55.0

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof

      [Start] 55.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 55.0	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 55.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 55.0	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 55.0	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 55.0	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 55.0	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 55.0	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 48.8

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot -0.5}}} \]

   4. Taylor expanded in b around -inf 64.0

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} + 4 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}}} \]

   5. Simplified 6.0

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

      [Start] 64.0	\[ \frac{1}{\frac{a}{b} + 4 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}} \]
      associate-*r/ [=>] 64.0	\[ \frac{1}{\frac{a}{b} + \color{blue}{\frac{4 \cdot b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}}} \]
      *-commutative [=>] 64.0	\[ \frac{1}{\frac{a}{b} + \frac{4 \cdot b}{\color{blue}{{\left(\sqrt{-4}\right)}^{2} \cdot c}}} \]
      times-frac [=>] 64.0	\[ \frac{1}{\frac{a}{b} + \color{blue}{\frac{4}{{\left(\sqrt{-4}\right)}^{2}} \cdot \frac{b}{c}}} \]
      unpow2 [=>] 64.0	\[ \frac{1}{\frac{a}{b} + \frac{4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c}} \]
      rem-square-sqrt [=>] 6.0	\[ \frac{1}{\frac{a}{b} + \frac{4}{\color{blue}{-4}} \cdot \frac{b}{c}} \]
      metadata-eval [=>] 6.0	\[ \frac{1}{\frac{a}{b} + \color{blue}{-1} \cdot \frac{b}{c}} \]

   if -6.40000000000000023e-9 < b < 1.62000000000000006e142

   1. Initial program 15.5

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

   if 1.62000000000000006e142 < b

   1. Initial program 57.9

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

   2. Simplified 57.9

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof

      [Start] 57.9	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 57.9	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 57.9	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 57.9	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 57.9	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 57.9	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 57.9	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 57.9	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around inf 2.0

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

   4. Simplified 2.0

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

      [Start] 2.0	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 2.0	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 2.0	\[ \frac{\color{blue}{-b}}{a} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 1.62 \cdot 10^{+142}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.7
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+141}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2
Error	14.3
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	22.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Error	39.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	22.4
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-260}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	56.4
Cost	192

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

Reproduce

herbie shell --seed 2023056 
(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)))