quadm (p42, negative)

Average Error: 34.3 → 10.1

Time: 18.1s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -5.2e-101)
   (/ -0.5 (fma -0.5 (/ a b) (* 0.5 (/ b c))))
   (if (<= b 1.45e+132)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 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 <= -5.2e-101) {
		tmp = -0.5 / fma(-0.5, (a / b), (0.5 * (b / c)));
	} else if (b <= 1.45e+132) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (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 <= -5.2e-101)
		tmp = Float64(-0.5 / fma(-0.5, Float64(a / b), Float64(0.5 * Float64(b / c))));
	elseif (b <= 1.45e+132)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / 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, -5.2e-101], N[(-0.5 / N[(-0.5 * N[(a / b), $MachinePrecision] + N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45e+132], 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[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	34.3
Target	20.8
Herbie	10.1

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

Derivation

1. Split input into 3 regimes

2. if b < -5.2000000000000002e-101

   1. Initial program 52.2

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

   2. Simplified 52.2

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

      [Start] 52.2	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 52.2	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 52.2	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 52.2	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 52.2	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 52.2	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 52.2	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 52.2	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 52.2	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 52.2	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 52.2	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]
      associate-*r* [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\left(-\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + b \cdot b}\right) \]
      distribute-lft-neg-in [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c} + b \cdot b}\right) \]
      *-commutative [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right)} + b \cdot b}\right) \]
      fma-def [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right) \]
      *-commutative [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, -\color{blue}{a \cdot 4}, b \cdot b\right)}\right) \]
      distribute-rgt-neg-in [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(-4\right)}, b \cdot b\right)}\right) \]
      metadata-eval [=>] 52.2	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-4}, b \cdot b\right)}\right) \]

   3. Applied egg-rr 52.2

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

   4. Applied egg-rr 46.8

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

   5. Taylor expanded in b around -inf 64.0

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

   6. Simplified 10.6

      \[\leadsto \frac{-0.5}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}} \]
      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 [=>] 10.6	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{-2}{\color{blue}{-4}} \cdot \frac{b}{c}\right)} \]
      metadata-eval [=>] 10.6	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{0.5} \cdot \frac{b}{c}\right)} \]

   if -5.2000000000000002e-101 < b < 1.4499999999999999e132

   1. Initial program 11.7

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

   if 1.4499999999999999e132 < b

   1. Initial program 56.7

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

   2. Simplified 56.7

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

      [Start] 56.7	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 56.7	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 56.7	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 56.7	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 56.7	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 56.7	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 56.7	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 56.7	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 56.7	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 56.7	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 56.7	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]
      associate-*r* [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\left(-\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + b \cdot b}\right) \]
      distribute-lft-neg-in [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c} + b \cdot b}\right) \]
      *-commutative [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right)} + b \cdot b}\right) \]
      fma-def [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right) \]
      *-commutative [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, -\color{blue}{a \cdot 4}, b \cdot b\right)}\right) \]
      distribute-rgt-neg-in [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(-4\right)}, b \cdot b\right)}\right) \]
      metadata-eval [=>] 56.7	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-4}, b \cdot b\right)}\right) \]

   3. Applied egg-rr 56.7

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

   4. Taylor expanded in b around inf 2.5

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

   5. Simplified 2.5

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

      [Start] 2.5	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 2.5	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 2.5	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-100}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+133}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{-100}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	13.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	13.4
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	40.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -15.5:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	22.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-252}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	56.5
Cost	192

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

Reproduce

herbie shell --seed 2022354 
(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)))