quadp (p42, positive)

Average Error: 34.4 → 9.6

Time: 15.8s

Precision: binary64

Cost: 14476

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+113}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{\left(c \cdot -4\right) \cdot \left(a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b - a \cdot \frac{c}{b}}{b \cdot \frac{-c}{b}}}\\ \end{array} \]

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 -1.6e+113)
   (/ (- b) a)
   (if (<= b 1.65e-247)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b 5e-45)
       (/
        (* (* c -4.0) (* a (/ 1.0 (+ b (hypot b (sqrt (* a (* c -4.0))))))))
        (* a 2.0))
       (/ 1.0 (/ (- b (* a (/ c b))) (* b (/ (- c) b))))))))

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 <= -1.6e+113) {
		tmp = -b / a;
	} else if (b <= 1.65e-247) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 5e-45) {
		tmp = ((c * -4.0) * (a * (1.0 / (b + hypot(b, sqrt((a * (c * -4.0)))))))) / (a * 2.0);
	} else {
		tmp = 1.0 / ((b - (a * (c / b))) / (b * (-c / b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+113) {
		tmp = -b / a;
	} else if (b <= 1.65e-247) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 5e-45) {
		tmp = ((c * -4.0) * (a * (1.0 / (b + Math.hypot(b, Math.sqrt((a * (c * -4.0)))))))) / (a * 2.0);
	} else {
		tmp = 1.0 / ((b - (a * (c / b))) / (b * (-c / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

↓

def code(a, b, c):
	tmp = 0
	if b <= -1.6e+113:
		tmp = -b / a
	elif b <= 1.65e-247:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	elif b <= 5e-45:
		tmp = ((c * -4.0) * (a * (1.0 / (b + math.hypot(b, math.sqrt((a * (c * -4.0)))))))) / (a * 2.0)
	else:
		tmp = 1.0 / ((b - (a * (c / b))) / (b * (-c / b)))
	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 <= -1.6e+113)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.65e-247)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= 5e-45)
		tmp = Float64(Float64(Float64(c * -4.0) * Float64(a * Float64(1.0 / Float64(b + hypot(b, sqrt(Float64(a * Float64(c * -4.0)))))))) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(b - Float64(a * Float64(c / b))) / Float64(b * Float64(Float64(-c) / b))));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.6e+113)
		tmp = -b / a;
	elseif (b <= 1.65e-247)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	elseif (b <= 5e-45)
		tmp = ((c * -4.0) * (a * (1.0 / (b + hypot(b, sqrt((a * (c * -4.0)))))))) / (a * 2.0);
	else
		tmp = 1.0 / ((b - (a * (c / b))) / (b * (-c / b)));
	end
	tmp_2 = 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, -1.6e+113], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.65e-247], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-45], N[(N[(N[(c * -4.0), $MachinePrecision] * N[(a * N[(1.0 / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b - N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[((-c) / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	34.4
Target	21.5
Herbie	9.6

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

Derivation

1. Split input into 4 regimes

2. if b < -1.5999999999999999e113

   1. Initial program 50.8

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

   2. Taylor expanded in b around -inf 3.0

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

   3. Simplified 3.0

      \[\leadsto \color{blue}{-\frac{b}{a}} \]
      Proof
      (neg.f64 (/.f64 b a)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 b a))): 0 points increase in error, 0 points decrease in error

   if -1.5999999999999999e113 < b < 1.64999999999999986e-247

   1. Initial program 10.2

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

   if 1.64999999999999986e-247 < b < 4.99999999999999976e-45

   1. Initial program 25.2

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

   2. Applied egg-rr 25.8

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

   3. Taylor expanded in b around 0 22.8

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

   4. Simplified 22.8

      \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(-4 \cdot c\right)}}{b + \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{2 \cdot a} \]
      Proof
      (*.f64 a (*.f64 -4 c)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -4 c) a)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -4 (*.f64 c a))): 2 points increase in error, 1 points decrease in error

   5. Applied egg-rr 17.2

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot c\right) \cdot \left(a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)}\right)}}{2 \cdot a} \]

   if 4.99999999999999976e-45 < b

   1. Initial program 54.1

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

   2. Simplified 54.1

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 4))))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) (neg.f64 4)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c))))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) 1)) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite<= metadata-eval (neg.f64 -1))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (neg.f64 -1)) (/.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite=> metadata-eval 1)) (/.f64 (/.f64 -1 2) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> /-rgt-identity_binary64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (/.f64 (/.f64 -1 2) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 2 a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) -1) (*.f64 2 a))): 13 points increase in error, 32 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 47.8

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

   4. Taylor expanded in b around inf 64.0

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} + 4 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}}} \]

   5. Simplified 8.7

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
      Proof
      (-.f64 (/.f64 a b) (/.f64 b c)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 a b) (neg.f64 (/.f64 b c)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a b) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 b c)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a b) (*.f64 (Rewrite<= metadata-eval (/.f64 4 -4)) (/.f64 b c))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a b) (*.f64 (/.f64 4 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -4) (sqrt.f64 -4)))) (/.f64 b c))): 194 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a b) (*.f64 (/.f64 4 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -4) 2))) (/.f64 b c))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a b) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 4 b) (*.f64 (pow.f64 (sqrt.f64 -4) 2) c)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a b) (/.f64 (*.f64 4 b) (Rewrite<= *-commutative_binary64 (*.f64 c (pow.f64 (sqrt.f64 -4) 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 a b) (Rewrite<= associate-*r/_binary64 (*.f64 4 (/.f64 b (*.f64 c (pow.f64 (sqrt.f64 -4) 2)))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 8.7

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(-a\right) \cdot \frac{c}{b} - \left(-b\right) \cdot 1}{\left(-b\right) \cdot \frac{c}{b}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+113}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{\left(c \cdot -4\right) \cdot \left(a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b - a \cdot \frac{c}{b}}{b \cdot \frac{-c}{b}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	62.4
Cost	192

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

Reproduce

herbie shell --seed 2022289 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))