quadm (p42, negative)

Average Error: 33.8 → 10.2

Time: 19.4s

Precision: binary64

Cost: 14152

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;\left(b + \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -2.05e-64)
   (/ (- c) b)
   (if (<= b 8.2e+116)
     (* (+ b (sqrt (/ 1.0 (/ 1.0 (fma b b (* a (* c -4.0))))))) (/ -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 <= -2.05e-64) {
		tmp = -c / b;
	} else if (b <= 8.2e+116) {
		tmp = (b + sqrt((1.0 / (1.0 / fma(b, b, (a * (c * -4.0))))))) * (-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 <= -2.05e-64)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 8.2e+116)
		tmp = Float64(Float64(b + sqrt(Float64(1.0 / Float64(1.0 / fma(b, b, Float64(a * Float64(c * -4.0))))))) * Float64(-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, -2.05e-64], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 8.2e+116], N[(N[(b + N[Sqrt[N[(1.0 / N[(1.0 / N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	33.8
Target	20.9
Herbie	10.2

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

Derivation

1. Split input into 3 regimes

2. if b < -2.05e-64

   1. Initial program 53.2

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

   2. Taylor expanded in b around -inf 8.8

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

   3. Simplified 8.8

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

      [Start] 8.8	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 8.8	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 8.8	\[ \frac{\color{blue}{-c}}{b} \]

   if -2.05e-64 < b < 8.1999999999999996e116

   1. Initial program 13.0

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

   2. Simplified 13.2

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

      [Start] 13.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 13.0	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 13.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 13.0	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 13.2	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 13.2	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 13.2	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 13.2	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 13.2

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

   4. Applied egg-rr 13.4

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

   if 8.1999999999999996e116 < b

   1. Initial program 51.4

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

   2. Simplified 51.4

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

      [Start] 51.4	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 51.4	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 51.4	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 51.4	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 51.5	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 51.5	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 51.5	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 51.5	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around inf 3.5

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

   4. Simplified 3.5

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

      [Start] 3.5	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 3.5	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 3.5	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;\left(b + \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.1
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.24 \cdot 10^{+117}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	13.8
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + {\left(\frac{-0.25}{c \cdot a}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + {\left(\frac{\frac{-0.25}{c}}{a}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Error	40.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+69}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	23.3
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-182}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	56.7
Cost	192

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

Reproduce

herbie shell --seed 2023039 
(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)))