quadp (p42, positive)

Average Error: 33.8 → 10.3

Time: 19.0s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+148}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -2e+148)
   (- (/ c b) (/ b a))
   (if (<= b 6.9e-125)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (/ (- 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 <= -2e+148) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.9e-125) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d+148)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.9d-125) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

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 <= -2e+148) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.9e-125) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = -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 <= -2e+148:
		tmp = (c / b) - (b / a)
	elif b <= 6.9e-125:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = -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 <= -2e+148)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.9e-125)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = 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 <= -2e+148)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.9e-125)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = -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, -2e+148], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.9e-125], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Target

Original	33.8
Target	20.8
Herbie	10.3

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

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e148

   1. Initial program 61.8

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

   2. Simplified 61.9

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}} \]
      Proof
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -4)))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 4)))))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) (neg.f64 4))))) b) (/.f64 1/2 a)): 16 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 4) (*.f64 a c))))) b) (/.f64 1/2 a)): 16 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 a c)))))) b) (/.f64 1/2 a)): 0 points increase in error, 16 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) b) (/.f64 1/2 a)): 16 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))) (neg.f64 b))) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (/.f64 (Rewrite<= metadata-eval (/.f64 1 2)) a)): 0 points increase in error, 16 points decrease in error
      (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (/.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) 2) a)): 6 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (/.f64 (neg.f64 -1) 2)) a)): 16 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l*_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (/.f64 a (/.f64 (neg.f64 -1) 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a 2) (neg.f64 -1)))): 0 points increase in error, 16 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 a)) (neg.f64 -1))): 16 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (/.f64 (*.f64 2 a) (Rewrite=> metadata-eval 1))): 0 points increase in error, 16 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite=> /-rgt-identity_binary64 (*.f64 2 a))): 0 points increase in error, 6 points decrease in error

   3. Applied egg-rr 37.0

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

   4. Applied egg-rr 37.0

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

   5. Taylor expanded in b around -inf 64.0

      \[\leadsto \color{blue}{-0.25 \cdot \frac{c \cdot {\left(\sqrt{-4}\right)}^{2}}{b} + -1 \cdot \frac{b}{a}} \]

   6. Simplified 2.6

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
      Proof
      (-.f64 (/.f64 c b) (/.f64 b a)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 c)) b) (/.f64 b a)): 3 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 (Rewrite<= metadata-eval (*.f64 -1/4 -4)) c) b) (/.f64 b a)): 0 points increase in error, 3 points decrease in error
      (-.f64 (/.f64 (*.f64 (*.f64 -1/4 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -4) (sqrt.f64 -4)))) c) b) (/.f64 b a)): 0 points increase in error, 3 points decrease in error
      (-.f64 (/.f64 (*.f64 (*.f64 -1/4 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -4) 2))) c) b) (/.f64 b a)): 3 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1/4 (*.f64 (pow.f64 (sqrt.f64 -4) 2) c))) b) (/.f64 b a)): 3 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 -1/4 (Rewrite<= *-commutative_binary64 (*.f64 c (pow.f64 (sqrt.f64 -4) 2)))) b) (/.f64 b a)): 0 points increase in error, 3 points decrease in error
      (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1/4 (/.f64 (*.f64 c (pow.f64 (sqrt.f64 -4) 2)) b))) (/.f64 b a)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 c (pow.f64 (sqrt.f64 -4) 2)) b)) (neg.f64 (/.f64 b a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/4 (/.f64 (*.f64 c (pow.f64 (sqrt.f64 -4) 2)) b)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 b a)))): 0 points increase in error, 0 points decrease in error

   if -2.0000000000000001e148 < b < 6.89999999999999973e-125

   1. Initial program 11.1

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

   if 6.89999999999999973e-125 < b

   1. Initial program 51.3

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

   2. Taylor expanded in b around inf 11.4

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

   3. Simplified 11.4

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Proof
      (/.f64 (neg.f64 c) b): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 c)) b): 0 points increase in error, 3 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 c b))): 0 points increase in error, 3 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+148}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.9 \cdot 10^{-125}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-88}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-88}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Error	40.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 3700000000000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 5
Error	22.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-210}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Error	56.7
Cost	192

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

Reproduce

herbie shell --seed 2022343 
(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)))