quadm (p42, negative)

Average Error: 34.2 → 10.3

Time: 20.8s

Precision: binary64

Cost: 13896

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+105}:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 -3e-90)
   (/ (- c) b)
   (if (<= b 8.5e+105)
     (* (+ b (sqrt (fma c (* a -4.0) (* b b)))) (/ -0.5 a))
     (/ (- 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 <= -3e-90) {
		tmp = -c / b;
	} else if (b <= 8.5e+105) {
		tmp = (b + sqrt(fma(c, (a * -4.0), (b * b)))) * (-0.5 / a);
	} 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 <= -3e-90)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 8.5e+105)
		tmp = Float64(Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return 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, -3e-90], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 8.5e+105], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Target

Original	34.2
Target	20.9
Herbie	10.3

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

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000002e-90

   1. Initial program 52.3

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

   2. Simplified 52.3

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

      [Start] 52.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 52.3	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 52.3	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 52.3	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 52.3	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 52.3	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 52.3	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 52.3	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 52.3	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 52.3	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 52.3	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 52.3	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 52.3	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 52.3	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 52.3	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Taylor expanded in b around -inf 10.3

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

   4. Simplified 10.3

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

      [Start] 10.3	\[ -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 10.3	\[ \color{blue}{-\frac{c}{b}} \]
      distribute-neg-frac [=>] 10.3	\[ \color{blue}{\frac{-c}{b}} \]

   if -3.0000000000000002e-90 < b < 8.49999999999999986e105

   1. Initial program 12.6

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

   2. Simplified 12.7

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

      [Start] 12.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 12.6	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 12.6	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 12.6	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 12.8	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 12.8	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 12.8	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 12.8	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 12.7	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 12.7	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 12.7	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 12.7	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 12.7	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 12.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 12.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   if 8.49999999999999986e105 < b

   1. Initial program 46.5

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

   2. Simplified 46.6

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

      [Start] 46.5	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 46.5	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 46.5	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 46.5	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 46.6	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 46.6	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 46.6	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 46.6	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 46.6	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 46.6	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 46.6	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 46.6	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 46.6	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 46.6	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 46.6	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Taylor expanded in a around 0 3.3

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

   4. Simplified 3.3

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

      [Start] 3.3	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 3.3	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 3.3	\[ \frac{\color{blue}{-b}}{a} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+105}:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-92}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2
Error	10.3
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+105}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3
Error	13.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-110}:\\ \;\;\;\;\left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4
Error	39.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Error	22.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-218}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	62.4
Cost	192

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

Alternative 7
Error	56.5
Cost	192

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

Reproduce

herbie shell --seed 2023187 
(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)))