quad2p (problem 3.2.1, positive)

Average Error: 34.0 → 10.2

Time: 16.0s

Precision: binary64

Cost: 7432

Math

\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{+63}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e+63)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.2e-69)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e+63) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.2e-69) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function

↓

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-9d+63)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 1.2d-69) then
        tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e+63) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.2e-69) {
		tmp = (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

↓

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e+63:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 1.2e-69:
		tmp = (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

↓

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e+63)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1.2e-69)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end

↓

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9e+63)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 1.2e-69)
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e+63], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.2e-69], N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.00000000000000034e63

   1. Initial program 40.7

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 5.5

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]

   3. Simplified 5.5

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
      Proof

      [Start] 5.5	\[ \frac{-2 \cdot b_2}{a} \]
      rational_best-simplify-2 [=>] 5.5	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if -9.00000000000000034e63 < b_2 < 1.2000000000000001e-69

   1. Initial program 13.6

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 1.2000000000000001e-69 < b_2

   1. Initial program 53.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 8.9

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{+63}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.3
Cost	7504

\[\begin{array}{l} t_0 := \sqrt{-\frac{c}{a}}\\ \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{-25}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -4 \cdot 10^{-32}:\\ \;\;\;\;-t_0\\ \mathbf{elif}\;b_2 \leq -1.25 \cdot 10^{-218}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.6 \cdot 10^{-145}:\\ \;\;\;\;-\left(t_0 + \frac{b_2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2
Error	13.7
Cost	7504

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{-25}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -4 \cdot 10^{-32}:\\ \;\;\;\;-\sqrt{-\frac{c}{a}}\\ \mathbf{elif}\;b_2 \leq -1.95 \cdot 10^{-52}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 4.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} + \left(-b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3
Error	20.4
Cost	7248

\[\begin{array}{l} t_0 := -\sqrt{-\frac{c}{a}}\\ \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{-18}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.2 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -1.25 \cdot 10^{-218}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 6 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4
Error	19.8
Cost	6920

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.9 \cdot 10^{-161}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 9.6 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Error	36.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 8.5 \cdot 10^{-204}:\\ \;\;\;\;-\frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6
Error	22.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 7
Error	59.1
Cost	256

\[-\frac{b_2}{a} \]

Reproduce

herbie shell --seed 2023094 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))