quadm (p42, negative)

Average Error: 53.94% → 17.68%

Time: 18.5s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 720000000:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}, \frac{-0.5}{a}, \frac{b \cdot -0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -1.75e-64)
   (/ (- c) b)
   (if (<= b 720000000.0)
     (fma (sqrt (fma c (* a -4.0) (* b b))) (/ -0.5 a) (/ (* b -0.5) a))
     (- (/ 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 <= -1.75e-64) {
		tmp = -c / b;
	} else if (b <= 720000000.0) {
		tmp = fma(sqrt(fma(c, (a * -4.0), (b * b))), (-0.5 / a), ((b * -0.5) / a));
	} 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 <= -1.75e-64)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 720000000.0)
		tmp = fma(sqrt(fma(c, Float64(a * -4.0), Float64(b * b))), Float64(-0.5 / a), Float64(Float64(b * -0.5) / a));
	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, -1.75e-64], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 720000000.0], N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.5 / a), $MachinePrecision] + N[(N[(b * -0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	53.94%
Target	33.1%
Herbie	17.68%

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

Derivation

1. Split input into 3 regimes

2. if b < -1.7500000000000002e-64

   1. Initial program 83.28

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

   2. Taylor expanded in b around -inf 13.85

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

   3. Simplified 13.85

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

      [Start] 13.85	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 13.85	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 13.85	\[ \frac{\color{blue}{-c}}{b} \]

   if -1.7500000000000002e-64 < b < 7.2e8

   1. Initial program 25.2

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

   2. Simplified 25.36

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

      [Start] 25.2	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 25.2	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 25.2	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 25.2	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 25.39	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 25.39	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 25.39	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 25.39	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 25.35	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 25.35	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 25.35	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 25.35	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 25.35	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 25.35	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 25.35	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Applied egg-rr 44.59

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left({\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}^{1.5}\right)}^{0.3333333333333333}}\right) \]

   4. Simplified 41.01

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

      [Start] 44.59	\[ \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \]
      unpow1/3 [=>] 41.01	\[ \frac{-0.5}{a} \cdot \left(b + \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}^{1.5}}}\right) \]

   5. Applied egg-rr 25.34

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

   6. Simplified 25.31

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

      [Start] 25.34	\[ \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \frac{-0.5}{a} + \frac{b}{a} \cdot -0.5 \]
      fma-def [=>] 25.31	\[ \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}, \frac{-0.5}{a}, \frac{b}{a} \cdot -0.5\right)} \]
      *-commutative [=>] 25.31	\[ \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}, \frac{-0.5}{a}, \color{blue}{-0.5 \cdot \frac{b}{a}}\right) \]
      associate-*r/ [=>] 25.31	\[ \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}, \frac{-0.5}{a}, \color{blue}{\frac{-0.5 \cdot b}{a}}\right) \]

   if 7.2e8 < b

   1. Initial program 52.32

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

   2. Simplified 52.55

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

      [Start] 52.32	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 52.32	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 52.32	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 52.32	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 52.49	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 52.49	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 52.49	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 52.49	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 52.44	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 52.44	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 52.44	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 52.44	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 52.44	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 52.44	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 52.44	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Applied egg-rr 52.54

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

   4. Taylor expanded in a around 0 11.55

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

   5. Simplified 11.55

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

      [Start] 11.55	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 11.55	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 11.55	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 17.68

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 720000000:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}, \frac{-0.5}{a}, \frac{b \cdot -0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.64%
Cost	7880

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 720000000:\\ \;\;\;\;-0.5 \cdot \frac{b}{a} + \frac{-0.5}{a} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	17.59%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-65}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 720000000:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	17.65%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 660000000:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	22.64%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -2.85 \cdot 10^{-65}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	22.65%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.82 \cdot 10^{-65}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6
Error	35.21%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 7
Error	62.39%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 8
Error	35.36%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-207}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 9
Error	97.39%
Cost	192

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

Alternative 10
Error	88.42%
Cost	192

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

Reproduce

herbie shell --seed 2023115 
(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)))