quad2p (problem 3.2.1, positive)

Average Error: 53.27% → 15.84%

Time: 14.2s

Precision: binary64

Cost: 7496

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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 -2e+150)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.65e-66)
     (- (/ (sqrt (- (* b_2 b_2) (* a c))) a) (/ b_2 a))
     (/ (* c -0.5) 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 <= -2e+150) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.65e-66) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) / a) - (b_2 / a);
	} else {
		tmp = (c * -0.5) / 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 <= (-2d+150)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 1.65d-66) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) / a) - (b_2 / a)
    else
        tmp = (c * (-0.5d0)) / 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 <= -2e+150) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.65e-66) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) / a) - (b_2 / a);
	} else {
		tmp = (c * -0.5) / 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 <= -2e+150:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 1.65e-66:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) / a) - (b_2 / a)
	else:
		tmp = (c * -0.5) / 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 <= -2e+150)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1.65e-66)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) / a) - Float64(b_2 / a));
	else
		tmp = Float64(Float64(c * -0.5) / 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 <= -2e+150)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 1.65e-66)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) / a) - (b_2 / a);
	else
		tmp = (c * -0.5) / 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, -2e+150], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.65e-66], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.99999999999999996e150

   1. Initial program 98.63

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

   2. Simplified 98.63

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

      [Start] 98.63	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 98.63	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 98.63	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 4.04

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

   4. Simplified 4.04

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

      [Start] 4.04	\[ \frac{-2 \cdot b_2}{a} \]
      *-commutative [=>] 4.04	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if -1.99999999999999996e150 < b_2 < 1.6499999999999999e-66

   1. Initial program 20.26

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

   2. Simplified 20.26

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

      [Start] 20.26	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 20.26	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 20.26	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 20.26

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

   if 1.6499999999999999e-66 < b_2

   1. Initial program 83.29

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

   2. Simplified 83.29

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

      [Start] 83.29	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 83.29	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 83.29	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 83.29

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

   4. Taylor expanded in b_2 around inf 13.45

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   5. Simplified 13.44

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
      Proof

      [Start] 13.45	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 13.44	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      *-commutative [=>] 13.44	\[ \frac{\color{blue}{c \cdot -0.5}}{b_2} \]

3. Recombined 3 regimes into one program.

4. Final simplification 15.84

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.07%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7.8 \cdot 10^{+84}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2
Error	30.08%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-118}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.55 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\frac{-c}{a}} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	21.51%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4
Error	34.86%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.9 \cdot 10^{-188}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Error	34.75%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.7 \cdot 10^{-188}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6
Error	34.74%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.4 \cdot 10^{-188}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 7
Error	61.94%
Cost	320

\[-0.5 \cdot \frac{c}{b_2} \]

Reproduce

herbie shell --seed 2023089 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))