quad2m (problem 3.2.1, negative)

Average Accuracy: 46.6% → 84.2%

Time: 14.7s

Precision: binary64

Cost: 7432

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \mathbf{elif}\;b_2 \leq 6 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

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 -1.1e-44)
   (/ c (/ b_2 -0.5))
   (if (<= b_2 6e+88)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 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 <= -1.1e-44) {
		tmp = c / (b_2 / -0.5);
	} else if (b_2 <= 6e+88) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (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 <= (-1.1d-44)) then
        tmp = c / (b_2 / (-0.5d0))
    else if (b_2 <= 6d+88) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (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 <= -1.1e-44) {
		tmp = c / (b_2 / -0.5);
	} else if (b_2 <= 6e+88) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (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 <= -1.1e-44:
		tmp = c / (b_2 / -0.5)
	elif b_2 <= 6e+88:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (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 <= -1.1e-44)
		tmp = Float64(c / Float64(b_2 / -0.5));
	elseif (b_2 <= 6e+88)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + 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 <= -1.1e-44)
		tmp = c / (b_2 / -0.5);
	elseif (b_2 <= 6e+88)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (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, -1.1e-44], N[(c / N[(b$95$2 / -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6e+88], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.10000000000000006e-44

   1. Initial program 15.5%

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

   2. Taylor expanded in b_2 around -inf 69.3%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}} + -0.5 \cdot \frac{c}{b_2}} \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{c \cdot c}{\frac{{b_2}^{3}}{a}}, -0.5 \cdot \frac{c}{b_2}\right)} \]
      Proof

      [Start] 69.3	\[ -0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}} + -0.5 \cdot \frac{c}{b_2} \]
      fma-def [=>] 69.3	\[ \color{blue}{\mathsf{fma}\left(-0.125, \frac{{c}^{2} \cdot a}{{b_2}^{3}}, -0.5 \cdot \frac{c}{b_2}\right)} \]
      associate-/l* [=>] 72.2	\[ \mathsf{fma}\left(-0.125, \color{blue}{\frac{{c}^{2}}{\frac{{b_2}^{3}}{a}}}, -0.5 \cdot \frac{c}{b_2}\right) \]
      unpow2 [=>] 72.2	\[ \mathsf{fma}\left(-0.125, \frac{\color{blue}{c \cdot c}}{\frac{{b_2}^{3}}{a}}, -0.5 \cdot \frac{c}{b_2}\right) \]

   4. Taylor expanded in c around 0 87.7%

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

   5. Simplified 87.6%

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

      [Start] 87.7	\[ -0.5 \cdot \frac{c}{b_2} \]
      *-commutative [=>] 87.7	\[ \color{blue}{\frac{c}{b_2} \cdot -0.5} \]
      associate-/r/ [<=] 87.6	\[ \color{blue}{\frac{c}{\frac{b_2}{-0.5}}} \]

   if -1.10000000000000006e-44 < b_2 < 6.00000000000000011e88

   1. Initial program 78.0%

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

   if 6.00000000000000011e88 < b_2

   1. Initial program 28.6%

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

   2. Taylor expanded in b_2 around inf 93.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \mathbf{elif}\;b_2 \leq 6 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	79.0%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \mathbf{elif}\;b_2 \leq 3.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2
Accuracy	64.9%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3
Accuracy	38.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 4
Accuracy	64.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-216}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 5
Accuracy	64.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.7 \cdot 10^{-216}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 6
Accuracy	11.5%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023133 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))