quadp (p42, positive)

Average Error: 34.1 → 10.3

Time: 14.7s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

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 -9.5e+60)
   (- (/ c b) (/ b a))
   (if (<= b 2.5e-50)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (+ a 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 <= -9.5e+60) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.5e-50) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a + 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 <= (-9.5d+60)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.5d-50) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a + 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 <= -9.5e+60) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.5e-50) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a + 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 <= -9.5e+60:
		tmp = (c / b) - (b / a)
	elif b <= 2.5e-50:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a + 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 <= -9.5e+60)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.5e-50)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a + 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 <= -9.5e+60)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.5e-50)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a + 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, -9.5e+60], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e-50], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Target

Original	34.1
Target	20.9
Herbie	10.3

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

Derivation

1. Split input into 3 regimes

2. if b < -9.49999999999999988e60

   1. Initial program 40.6

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

   2. Simplified 40.6

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

      [Start] 40.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 40.6	\[ \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \color{blue}{\left(0 - b\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 40.6	\[ \frac{\color{blue}{\left(0 + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) - b}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 40.6	\[ \frac{\color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + 0\right)} - b}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 40.6	\[ \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{a \cdot 2}} \]
      metadata-eval [<=] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \color{blue}{\left(1 + 1\right)}} \]
      metadata-eval [<=] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \left(\color{blue}{\frac{4}{4}} + 1\right)} \]
      metadata-eval [<=] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \left(\frac{4}{4} + \color{blue}{\frac{4}{4}}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [<=] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{\frac{4}{4} \cdot a + a \cdot \frac{4}{4}}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{a \cdot \frac{4}{4}} + a \cdot \frac{4}{4}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \frac{4}{4} + \color{blue}{\frac{4}{4} \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{\frac{4}{4} \cdot \left(a + a\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{\left(a + a\right) \cdot \frac{4}{4}}} \]
      metadata-eval [=>] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\left(a + a\right) \cdot \color{blue}{1}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 40.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{a + a}} \]

   3. Taylor expanded in b around -inf 5.6

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

   4. Simplified 5.6

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

      [Start] 5.6	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 5.6	\[ \frac{c}{b} + \color{blue}{\frac{b}{a} \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 5.6	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

   5. Applied egg-rr 5.6

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

   if -9.49999999999999988e60 < b < 2.49999999999999984e-50

   1. Initial program 14.2

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

   2. Simplified 14.2

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

      [Start] 14.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 14.2	\[ \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \color{blue}{\left(0 - b\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 14.2	\[ \frac{\color{blue}{\left(0 + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) - b}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 14.2	\[ \frac{\color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + 0\right)} - b}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 14.2	\[ \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{a \cdot 2}} \]
      metadata-eval [<=] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \color{blue}{\left(1 + 1\right)}} \]
      metadata-eval [<=] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \left(\color{blue}{\frac{4}{4}} + 1\right)} \]
      metadata-eval [<=] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \left(\frac{4}{4} + \color{blue}{\frac{4}{4}}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [<=] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{\frac{4}{4} \cdot a + a \cdot \frac{4}{4}}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{a \cdot \frac{4}{4}} + a \cdot \frac{4}{4}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \frac{4}{4} + \color{blue}{\frac{4}{4} \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{\frac{4}{4} \cdot \left(a + a\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{\left(a + a\right) \cdot \frac{4}{4}}} \]
      metadata-eval [=>] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\left(a + a\right) \cdot \color{blue}{1}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 14.2	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{a + a}} \]

   if 2.49999999999999984e-50 < b

   1. Initial program 54.5

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

   2. Simplified 54.5

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

      [Start] 54.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 54.5	\[ \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \color{blue}{\left(0 - b\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 54.5	\[ \frac{\color{blue}{\left(0 + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) - b}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 54.5	\[ \frac{\color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + 0\right)} - b}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 54.5	\[ \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{a \cdot 2}} \]
      metadata-eval [<=] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \color{blue}{\left(1 + 1\right)}} \]
      metadata-eval [<=] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \left(\color{blue}{\frac{4}{4}} + 1\right)} \]
      metadata-eval [<=] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \left(\frac{4}{4} + \color{blue}{\frac{4}{4}}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [<=] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{\frac{4}{4} \cdot a + a \cdot \frac{4}{4}}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{a \cdot \frac{4}{4}} + a \cdot \frac{4}{4}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot \frac{4}{4} + \color{blue}{\frac{4}{4} \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{\frac{4}{4} \cdot \left(a + a\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{\left(a + a\right) \cdot \frac{4}{4}}} \]
      metadata-eval [=>] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\left(a + a\right) \cdot \color{blue}{1}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 54.5	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\color{blue}{a + a}} \]

   3. Taylor expanded in b around inf 8.0

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

   4. Simplified 8.0

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

      [Start] 8.0	\[ -1 \cdot \frac{c}{b} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 8.0	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 8.0	\[ \color{blue}{-\frac{c}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 2
Error	40.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{+75}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 3
Error	22.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-214}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 4
Error	56.9
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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)))