quad2p (problem 3.2.1, positive)

Average Error: 34.0 → 10.5

Time: 14.4s

Precision: binary64

Cost: 7432

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.28 \cdot 10^{-123}:\\ \;\;\;\;\frac{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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 -9.4e+114)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.28e-123)
     (/ (- (pow (- (* b_2 b_2) (* a c)) 0.5) 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 <= -9.4e+114) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.28e-123) {
		tmp = (pow(((b_2 * b_2) - (a * c)), 0.5) - 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 <= (-9.4d+114)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 1.28d-123) then
        tmp = ((((b_2 * b_2) - (a * c)) ** 0.5d0) - 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 <= -9.4e+114) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.28e-123) {
		tmp = (Math.pow(((b_2 * b_2) - (a * c)), 0.5) - 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 <= -9.4e+114:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 1.28e-123:
		tmp = (math.pow(((b_2 * b_2) - (a * c)), 0.5) - 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 <= -9.4e+114)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1.28e-123)
		tmp = Float64(Float64((Float64(Float64(b_2 * b_2) - Float64(a * c)) ^ 0.5) - 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 <= -9.4e+114)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 1.28e-123)
		tmp = ((((b_2 * b_2) - (a * c)) ^ 0.5) - 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, -9.4e+114], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.28e-123], N[(N[(N[Power[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.4000000000000001e114

   1. Initial program 49.2

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

   2. Simplified 49.2

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

      [Start] 49.2	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 49.2	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 49.2	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 3.6

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

   4. Simplified 3.6

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

      [Start] 3.6	\[ \frac{-2 \cdot b_2}{a} \]
      *-commutative [=>] 3.6	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if -9.4000000000000001e114 < b_2 < 1.28000000000000002e-123

   1. Initial program 11.6

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

   2. Simplified 11.6

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

      [Start] 11.6	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 11.6	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 11.6	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 11.6

      \[\leadsto \frac{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}} - b_2}{a} \]

   if 1.28000000000000002e-123 < b_2

   1. Initial program 51.4

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

   2. Simplified 51.4

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

      [Start] 51.4	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 51.4	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 51.4	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 11.7

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

   4. Simplified 11.7

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

      [Start] 11.7	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 11.7	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      *-commutative [=>] 11.7	\[ \frac{\color{blue}{c \cdot -0.5}}{b_2} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.28 \cdot 10^{-123}:\\ \;\;\;\;\frac{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.45 \cdot 10^{+115}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.28 \cdot 10^{-123}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{-95}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.35 \cdot 10^{-131}:\\ \;\;\;\;\frac{{\left(c \cdot \left(-a\right)\right)}^{0.5} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	13.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{-95}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.28 \cdot 10^{-123}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4
Error	23.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.15 \cdot 10^{-194}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 5
Error	23.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.15 \cdot 10^{-194}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 6
Error	22.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.15 \cdot 10^{-194}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 7
Error	45.6
Cost	320

\[b_2 \cdot \frac{-2}{a} \]

Alternative 8
Error	45.6
Cost	320

\[\frac{-2}{\frac{a}{b_2}} \]

Alternative 9
Error	59.3
Cost	256

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

Reproduce

herbie shell --seed 2023002 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))