The quadratic formula (r2)

Average Error: 33.7 → 10.2

Time: 13.0s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 -6e-64)
   (/ (- c) b)
   (if (<= b 4.7e+83)
     (/ (- (- b) (sqrt (+ (* b b) (* (* c a) -4.0)))) (* 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 <= -6e-64) {
		tmp = -c / b;
	} else if (b <= 4.7e+83) {
		tmp = (-b - sqrt(((b * b) + ((c * a) * -4.0)))) / (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 <= (-6d-64)) then
        tmp = -c / b
    else if (b <= 4.7d+83) then
        tmp = (-b - sqrt(((b * b) + ((c * a) * (-4.0d0))))) / (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 <= -6e-64) {
		tmp = -c / b;
	} else if (b <= 4.7e+83) {
		tmp = (-b - Math.sqrt(((b * b) + ((c * a) * -4.0)))) / (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 <= -6e-64:
		tmp = -c / b
	elif b <= 4.7e+83:
		tmp = (-b - math.sqrt(((b * b) + ((c * a) * -4.0)))) / (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 <= -6e-64)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 4.7e+83)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + Float64(Float64(c * a) * -4.0)))) / 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 <= -6e-64)
		tmp = -c / b;
	elseif (b <= 4.7e+83)
		tmp = (-b - sqrt(((b * b) + ((c * a) * -4.0)))) / (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, -6e-64], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 4.7e+83], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Target

Original	33.7
Target	21.1
Herbie	10.2

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

Derivation

1. Split input into 3 regimes

2. if b < -6.0000000000000001e-64

   1. Initial program 53.4

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

   2. Simplified 53.4

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

   3. Taylor expanded in b around -inf 9.2

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

   4. Simplified 9.2

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      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, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 c b))): 0 points increase in error, 0 points decrease in error

   if -6.0000000000000001e-64 < b < 4.6999999999999999e83

   1. Initial program 13.1

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

   if 4.6999999999999999e83 < b

   1. Initial program 43.8

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

   2. Simplified 43.9

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)} \]
      Proof
      (*.f64 (/.f64 -1/2 a) (+.f64 b (sqrt.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) a) (+.f64 b (sqrt.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 2 a))) (+.f64 b (sqrt.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (fma.f64 (*.f64 a c) (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 a c) (neg.f64 4)) (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4))) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c)))) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (neg.f64 (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite<= sub-neg_binary64 (-.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite=> sub-neg_binary64 (+.f64 b (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (Rewrite=> remove-double-neg_binary64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (*.f64 2 a))): 4 points increase in error, 20 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. Taylor expanded in a around 0 4.6

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

   4. Simplified 4.6

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
      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 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+83}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2
Error	13.9
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-62}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	13.9
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{\left(c \cdot a\right) \cdot -4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	13.9
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-60}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-127}:\\ \;\;\;\;\frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	23.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{-304}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6
Error	39.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	23.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-286}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8
Error	56.8
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022338 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))