quad2p (problem 3.2.1, positive)

Average Error: 34.1 → 10.4

Time: 12.9s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{+70}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.55 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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 -1e+70)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 2.55e-87)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) 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 <= -1e+70) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.55e-87) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= (-1d+70)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 2.55d-87) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= -1e+70) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.55e-87) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - 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 <= -1e+70:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 2.55e-87:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - 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 <= -1e+70)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 2.55e-87)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - 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 <= -1e+70)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 2.55e-87)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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, -1e+70], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.55e-87], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $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 < -1.00000000000000007e70

   1. Initial program 40.8

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

   2. Simplified 40.8

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b_2 around -inf 4.6

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

   if -1.00000000000000007e70 < b_2 < 2.55000000000000012e-87

   1. Initial program 13.1

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

   2. Simplified 13.1

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 0 points increase in error, 0 points decrease in error

   if 2.55000000000000012e-87 < b_2

   1. Initial program 52.5

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

   2. Simplified 52.5

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b_2 around inf 10.1

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

   4. Simplified 10.1

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
      Proof
      (/.f64 (*.f64 c -1/2) b_2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 c b_2) -1/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (/.f64 c b_2))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{+70}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.55 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.7 \cdot 10^{-98}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 8.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2
Error	13.7
Cost	7048

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-132}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	39.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.9 \cdot 10^{-281}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	22.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.3 \cdot 10^{-195}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]

Alternative 5
Error	22.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.55 \cdot 10^{-195}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 6
Error	53.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.9 \cdot 10^{-281}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	56.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022329 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))