quadm (p42, negative)

Average Error: 34.1 → 9.9

Time: 16.7s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 -3e-118)
   (/ 1.0 (- (/ a b) (/ b c)))
   (if (<= b 2.5e+135)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* a 2.0))
     (/ (- 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 <= -3e-118) {
		tmp = 1.0 / ((a / b) - (b / c));
	} else if (b <= 2.5e+135) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	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 <= (-3d-118)) then
        tmp = 1.0d0 / ((a / b) - (b / c))
    else if (b <= 2.5d+135) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (a * 2.0d0)
    else
        tmp = -b / a
    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 <= -3e-118) {
		tmp = 1.0 / ((a / b) - (b / c));
	} else if (b <= 2.5e+135) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	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 <= -3e-118:
		tmp = 1.0 / ((a / b) - (b / c))
	elif b <= 2.5e+135:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0)
	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 <= -3e-118)
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	elseif (b <= 2.5e+135)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	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 <= -3e-118)
		tmp = 1.0 / ((a / b) - (b / c));
	elseif (b <= 2.5e+135)
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	else
		tmp = -b / a;
	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, -3e-118], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e+135], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Target

Original	34.1
Target	20.8
Herbie	9.9

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

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000018e-118

   1. Initial program 52.1

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

   2. Simplified 52.1

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      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)): 1 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)): 1 points increase in error, 1 points decrease in error
      (*.f64 (+.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, 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))): 8 points increase in error, 23 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<= distribute-neg-out_binary64 (+.f64 (neg.f64 b) (neg.f64 (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<= sub-neg_binary64 (-.f64 (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 46.9

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

   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 11.2

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
      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))): 200 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

   if -3.00000000000000018e-118 < b < 2.50000000000000015e135

   1. Initial program 10.7

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

   if 2.50000000000000015e135 < b

   1. Initial program 57.1

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

   2. Taylor expanded in b around inf 2.7

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

   3. Simplified 2.7

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
      Proof
      (/.f64 (neg.f64 b) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 b)) a): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 b a))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.1
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 3.95 \cdot 10^{+59}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(a \cdot c\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	14.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	13.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	22.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	22.4
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.36 \cdot 10^{-227}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	39.7
Cost	256

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

Alternative 8
Error	56.8
Cost	192

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

Reproduce

herbie shell --seed 2022300 
(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)))