quadm (p42, negative)

Average Error: 34.0 → 11.0

Time: 18.5s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -5.5e-150)
   (/ -0.5 (fma -0.5 (/ a b) (* 0.5 (/ b c))))
   (if (<= b 1.05e+79)
     (/ (- (- b) (sqrt (+ (* b b) (* (* a c) -4.0)))) (* 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 <= -5.5e-150) {
		tmp = -0.5 / fma(-0.5, (a / b), (0.5 * (b / c)));
	} else if (b <= 1.05e+79) {
		tmp = (-b - sqrt(((b * b) + ((a * c) * -4.0)))) / (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 <= -5.5e-150)
		tmp = Float64(-0.5 / fma(-0.5, Float64(a / b), Float64(0.5 * Float64(b / c))));
	elseif (b <= 1.05e+79)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - 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, -5.5e-150], N[(-0.5 / N[(-0.5 * N[(a / b), $MachinePrecision] + N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+79], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	34.0
Target	21.0
Herbie	11.0

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

Derivation

1. Split input into 3 regimes

2. if b < -5.4999999999999996e-150

   1. Initial program 50.0

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

   2. Simplified 50.0

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

      [Start] 50.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 50.0	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 50.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 50.0	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 50.0	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 50.0	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 50.0	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 50.0	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 44.8

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

   4. Taylor expanded in b around -inf 64.0

      \[\leadsto \frac{-0.5}{\color{blue}{-2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}} + -0.5 \cdot \frac{a}{b}}} \]

   5. Simplified 12.6

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

      [Start] 64.0	\[ \frac{-0.5}{-2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}} + -0.5 \cdot \frac{a}{b}} \]
      +-commutative [=>] 64.0	\[ \frac{-0.5}{\color{blue}{-0.5 \cdot \frac{a}{b} + -2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}}} \]
      fma-def [=>] 64.0	\[ \frac{-0.5}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{a}{b}, -2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}} \]
      associate-*r/ [=>] 64.0	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{\frac{-2 \cdot b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}}\right)} \]
      *-commutative [=>] 64.0	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{-2 \cdot b}{\color{blue}{{\left(\sqrt{-4}\right)}^{2} \cdot c}}\right)} \]
      times-frac [=>] 64.0	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{\frac{-2}{{\left(\sqrt{-4}\right)}^{2}} \cdot \frac{b}{c}}\right)} \]
      unpow2 [=>] 64.0	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{-2}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c}\right)} \]
      rem-square-sqrt [=>] 12.6	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{-2}{\color{blue}{-4}} \cdot \frac{b}{c}\right)} \]
      metadata-eval [=>] 12.6	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{0.5} \cdot \frac{b}{c}\right)} \]

   if -5.4999999999999996e-150 < b < 1.05000000000000004e79

   1. Initial program 11.9

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

   if 1.05000000000000004e79 < b

   1. Initial program 43.1

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

   2. Simplified 43.2

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

      [Start] 43.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 43.1	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 43.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 43.1	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 43.2	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 43.2	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 43.2	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 43.2	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 57.9

      \[\leadsto \left(b + \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5}}}\right) \cdot \frac{-0.5}{a} \]

   4. Taylor expanded in b around inf 4.4

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

   5. Simplified 4.4

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

      [Start] 4.4	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 4.4	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 4.4	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.0
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+78}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	14.2
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3
Error	14.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-149}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Error	39.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	22.7
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-214}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	62.3
Cost	192

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

Alternative 8
Error	56.6
Cost	192

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

Reproduce

herbie shell --seed 2023034 
(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)))