quadm (p42, negative)

Average Error: 34.1 → 10.5

Time: 7.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -6.6e-33)
   (/ (- c) b)
   (if (<= b 6.5e+109)
     (/ (- (- b) (sqrt (fma b b (* a (* c -4.0))))) (* 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 <= -6.6e-33) {
		tmp = -c / b;
	} else if (b <= 6.5e+109) {
		tmp = (-b - sqrt(fma(b, b, (a * (c * -4.0))))) / (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 <= -6.6e-33)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 6.5e+109)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(b, b, Float64(a * Float64(c * -4.0))))) / 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, -6.6e-33], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 6.5e+109], N[(N[((-b) - N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	34.1
Target	21.2
Herbie	10.5

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

Derivation

1. Split input into 3 regimes

2. if b < -6.6000000000000005e-33

   1. Initial program 54.1

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

   2. Taylor expanded in b around -inf 7.4

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

   3. Simplified 7.4

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

   if -6.6000000000000005e-33 < b < 6.5e109

   1. Initial program 14.9

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

   2. Taylor expanded in b around 0 14.9

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

   3. Simplified 15.0

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

   if 6.5e109 < b

   1. Initial program 48.9

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

   2. Taylor expanded in b around inf 3.4

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

   3. Simplified 3.4

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(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)))