quadm (p42, negative)

Average Error: 33.6 → 10.4

Time: 20.4s

Precision: binary64

Cost: 21064

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -0.135:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{t_0 - b \cdot b}{\sqrt{t_0} - b}\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+117}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b))) (t_1 (/ (- c) b)))
   (if (<= b -0.135)
     t_1
     (if (<= b -1.8e-28)
       (* (/ -0.5 a) (/ (- t_0 (* b b)) (- (sqrt t_0) b)))
       (if (<= b -8e-76)
         t_1
         (if (<= b 1.95e+117)
           (/ (- (- b) (sqrt (+ (* b b) (* -4.0 (* c a))))) (* 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 t_0 = fma(c, (a * -4.0), (b * b));
	double t_1 = -c / b;
	double tmp;
	if (b <= -0.135) {
		tmp = t_1;
	} else if (b <= -1.8e-28) {
		tmp = (-0.5 / a) * ((t_0 - (b * b)) / (sqrt(t_0) - b));
	} else if (b <= -8e-76) {
		tmp = t_1;
	} else if (b <= 1.95e+117) {
		tmp = (-b - sqrt(((b * b) + (-4.0 * (c * a))))) / (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)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	t_1 = Float64(Float64(-c) / b)
	tmp = 0.0
	if (b <= -0.135)
		tmp = t_1;
	elseif (b <= -1.8e-28)
		tmp = Float64(Float64(-0.5 / a) * Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) - b)));
	elseif (b <= -8e-76)
		tmp = t_1;
	elseif (b <= 1.95e+117)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a))))) / 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_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -0.135], t$95$1, If[LessEqual[b, -1.8e-28], N[(N[(-0.5 / a), $MachinePrecision] * N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8e-76], t$95$1, If[LessEqual[b, 1.95e+117], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	33.6
Target	20.7
Herbie	10.4

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

Derivation

1. Split input into 4 regimes

2. if b < -0.13500000000000001 or -1.7999999999999999e-28 < b < -7.99999999999999942e-76

   1. Initial program 53.4

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

   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 -0.13500000000000001 < b < -1.7999999999999999e-28

   1. Initial program 41.3

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

   2. Simplified 41.4

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

      [Start] 41.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 41.3	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 41.3	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 41.3	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 41.4	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 41.4	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 41.4	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 41.4	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 41.4	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 41.4	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 41.4	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 41.4	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 41.4	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 41.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 41.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Applied egg-rr 41.3

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

   if -7.99999999999999942e-76 < b < 1.94999999999999995e117

   1. Initial program 12.3

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

   if 1.94999999999999995e117 < b

   1. Initial program 51.8

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

   2. Simplified 51.9

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

      [Start] 51.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 51.8	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 51.8	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 51.8	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 51.8	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 51.8	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 51.8	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 51.8	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 51.8	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 51.8	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 51.8	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 51.8	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 51.8	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 51.8	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 51.8	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Applied egg-rr 51.9

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

   4. Taylor expanded in a around 0 3.0

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

   5. Simplified 3.0

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

      [Start] 3.0	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 3.0	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 3.0	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.135:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+117}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	7952

\[\begin{array}{l} t_0 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -0.000105:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.55 \cdot 10^{-26}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+117}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	10.4
Cost	7888

\[\begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := \frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{if}\;b \leq -0.0037:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.7 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	14.1
Cost	7696

\[\begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -0.000105:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + t_0\right)\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-110}:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	14.2
Cost	7632

\[\begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := \frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{if}\;b \leq -0.000105:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.45 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.3 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	22.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6
Error	39.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+19}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	22.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-205}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8
Error	62.3
Cost	192

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

Alternative 9
Error	56.9
Cost	192

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

Reproduce

herbie shell --seed 2023059 
(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)))