quadm (p42, negative)

Average Error: 34.2 → 8.1

Time: 21.5s

Precision: binary64

Cost: 13832

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(\frac{c}{b} \cdot \left(a \cdot -4\right), -0.5, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-278}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+60}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 -7.5e-22)
   (/ (* c 2.0) (fma (* (/ c b) (* a -4.0)) -0.5 (* b -2.0)))
   (if (<= b 3.9e-278)
     (/ (* c 2.0) (- (hypot b (sqrt (* a (* c -4.0)))) b))
     (if (<= b 1.65e+60)
       (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* 2.0 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 <= -7.5e-22) {
		tmp = (c * 2.0) / fma(((c / b) * (a * -4.0)), -0.5, (b * -2.0));
	} else if (b <= 3.9e-278) {
		tmp = (c * 2.0) / (hypot(b, sqrt((a * (c * -4.0)))) - b);
	} else if (b <= 1.65e+60) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (2.0 * 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 <= -7.5e-22)
		tmp = Float64(Float64(c * 2.0) / fma(Float64(Float64(c / b) * Float64(a * -4.0)), -0.5, Float64(b * -2.0)));
	elseif (b <= 3.9e-278)
		tmp = Float64(Float64(c * 2.0) / Float64(hypot(b, sqrt(Float64(a * Float64(c * -4.0)))) - b));
	elseif (b <= 1.65e+60)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(2.0 * 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, -7.5e-22], N[(N[(c * 2.0), $MachinePrecision] / N[(N[(N[(c / b), $MachinePrecision] * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-278], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.65e+60], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]

Target

Original	34.2
Target	21.0
Herbie	8.1

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

Derivation

1. Split input into 4 regimes

2. if b < -7.49999999999999978e-22

   1. Initial program 55.2

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

   2. Simplified 55.2

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

      [Start] 55.2	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 55.2	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 55.2	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 55.2	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 55.2	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 55.2	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 55.2	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 55.2	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 57.0

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

   4. Taylor expanded in a around 0 29.2

      \[\leadsto \frac{\color{blue}{2 \cdot c}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b} \]

   5. Simplified 29.2

      \[\leadsto \frac{\color{blue}{c \cdot 2}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b} \]
      Proof

      [Start] 29.2	\[ \frac{2 \cdot c}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b} \]
      *-commutative [=>] 29.2	\[ \frac{\color{blue}{c \cdot 2}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b} \]

   6. Taylor expanded in b around -inf 64.0

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

   7. Simplified 6.0

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

      [Start] 64.0	\[ \frac{c \cdot 2}{-2 \cdot b + -0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}} \]
      +-commutative [=>] 64.0	\[ \frac{c \cdot 2}{\color{blue}{-0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b} + -2 \cdot b}} \]
      *-commutative [=>] 64.0	\[ \frac{c \cdot 2}{\color{blue}{\frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b} \cdot -0.5} + -2 \cdot b} \]
      fma-def [=>] 64.0	\[ \frac{c \cdot 2}{\color{blue}{\mathsf{fma}\left(\frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}, -0.5, -2 \cdot b\right)}} \]
      associate-/l* [=>] 64.0	\[ \frac{c \cdot 2}{\mathsf{fma}\left(\color{blue}{\frac{c}{\frac{b}{a \cdot {\left(\sqrt{-4}\right)}^{2}}}}, -0.5, -2 \cdot b\right)} \]
      associate-/r/ [=>] 64.0	\[ \frac{c \cdot 2}{\mathsf{fma}\left(\color{blue}{\frac{c}{b} \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}, -0.5, -2 \cdot b\right)} \]
      unpow2 [=>] 64.0	\[ \frac{c \cdot 2}{\mathsf{fma}\left(\frac{c}{b} \cdot \left(a \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}\right), -0.5, -2 \cdot b\right)} \]
      rem-square-sqrt [=>] 6.0	\[ \frac{c \cdot 2}{\mathsf{fma}\left(\frac{c}{b} \cdot \left(a \cdot \color{blue}{-4}\right), -0.5, -2 \cdot b\right)} \]
      *-commutative [=>] 6.0	\[ \frac{c \cdot 2}{\mathsf{fma}\left(\frac{c}{b} \cdot \left(a \cdot -4\right), -0.5, \color{blue}{b \cdot -2}\right)} \]

   if -7.49999999999999978e-22 < b < 3.9000000000000001e-278

   1. Initial program 24.5

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

   2. Simplified 24.6

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

      [Start] 24.5	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 24.5	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 24.5	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 24.5	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 24.6	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 24.6	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 24.6	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 24.6	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 25.0

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

   4. Taylor expanded in a around 0 14.2

      \[\leadsto \frac{\color{blue}{2 \cdot c}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b} \]

   5. Simplified 14.2

      \[\leadsto \frac{\color{blue}{c \cdot 2}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b} \]
      Proof

      [Start] 14.2	\[ \frac{2 \cdot c}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b} \]
      *-commutative [=>] 14.2	\[ \frac{\color{blue}{c \cdot 2}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b} \]

   if 3.9000000000000001e-278 < b < 1.6499999999999999e60

   1. Initial program 8.6

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

   if 1.6499999999999999e60 < b

   1. Initial program 40.0

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

   2. Simplified 40.0

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

      [Start] 40.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-rgt-identity [<=] 40.0	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot 1} \]
      metadata-eval [<=] 40.0	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \cdot \color{blue}{\left(--1\right)} \]
      associate-*l/ [=>] 40.0	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 40.1	\[ \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      distribute-neg-frac [<=] 40.1	\[ \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \]
      distribute-rgt-neg-in [<=] 40.1	\[ \color{blue}{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]
      distribute-lft-neg-out [<=] 40.1	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around inf 4.9

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

   4. Simplified 4.9

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

      [Start] 4.9	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 4.9	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 4.9	\[ \frac{\color{blue}{-b}}{a} \]

3. Recombined 4 regimes into one program.

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(\frac{c}{b} \cdot \left(a \cdot -4\right), -0.5, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-278}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+60}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-67}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(\frac{c}{b} \cdot \left(a \cdot -4\right), -0.5, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+60}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2
Error	9.9
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-63}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+60}:\\ \;\;\;\;\left(b + \sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(\frac{c}{b} \cdot \left(a \cdot -4\right), -0.5, b \cdot -2\right)}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+60}:\\ \;\;\;\;\left(b + \sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4
Error	12.9
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Error	12.9
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-62}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	39.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -14.5:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	22.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-248}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8
Error	56.7
Cost	192

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

Reproduce

herbie shell --seed 2023017 
(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)))