quadp (p42, positive)

Average Error: 34.4 → 10.6

Time: 17.3s

Precision: binary64

Cost: 7688

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.75 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.75e+153)
   (+ (/ c b) (- (/ b a)))
   (if (<= b 8.6e-147)
     (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.75e+153) {
		tmp = (c / b) + -(b / a);
	} else if (b <= 8.6e-147) {
		tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.75d+153)) then
        tmp = (c / b) + -(b / a)
    else if (b <= 8.6d-147) then
        tmp = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.75e+153) {
		tmp = (c / b) + -(b / a);
	} else if (b <= 8.6e-147) {
		tmp = (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

↓

def code(a, b, c):
	tmp = 0
	if b <= -2.75e+153:
		tmp = (c / b) + -(b / a)
	elif b <= 8.6e-147:
		tmp = (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.75e+153)
		tmp = Float64(Float64(c / b) + Float64(-Float64(b / a)));
	elseif (b <= 8.6e-147)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.75e+153)
		tmp = (c / b) + -(b / a);
	elseif (b <= 8.6e-147)
		tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := If[LessEqual[b, -2.75e+153], N[(N[(c / b), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 8.6e-147], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Target

Original	34.4
Target	21.5
Herbie	10.6

\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if b < -2.7500000000000001e153

   1. Initial program 63.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Simplified 63.9

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
      Proof

      [Start] 63.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 63.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 2.5

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

   4. Simplified 2.5

      \[\leadsto \color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)} \]
      Proof

      [Start] 2.5	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational.json-simplify-2 [=>] 2.5	\[ \frac{c}{b} + \color{blue}{\frac{b}{a} \cdot -1} \]
      rational.json-simplify-9 [=>] 2.5	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

   if -2.7500000000000001e153 < b < 8.6000000000000002e-147

   1. Initial program 10.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   if 8.6000000000000002e-147 < b

   1. Initial program 49.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Simplified 49.9

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
      Proof

      [Start] 49.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 49.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 12.7

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   4. Simplified 12.7

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

      [Start] 12.7	\[ -1 \cdot \frac{c}{b} \]
      rational.json-simplify-2 [=>] 12.7	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational.json-simplify-9 [=>] 12.7	\[ \color{blue}{-\frac{c}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.75 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.9
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 2
Error	10.9
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -6.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-147}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 3
Error	14.0
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-61}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-4 \cdot c\right)} + \left(-b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 4
Error	19.8
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ t_1 := -\frac{c}{b}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{-240}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	14.3
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-79}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{0.5}{a} \cdot \sqrt{\left(c \cdot a\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	39.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-10}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7
Error	21.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{-239}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 8
Error	56.8
Cost	192

\[\frac{c}{b} \]

Reproduce

herbie shell --seed 2023069 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))