quadp (p42, positive)

Average Error: 34.3 → 11.0

Time: 40.6s

Precision: binary64

Cost: 26892

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -8 \cdot 10^{+143}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \frac{{\left(e^{0.25 \cdot \left(\log c + \log \left(a \cdot -4\right)\right)}\right)}^{2} - b}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (+ (* b b) (* (* a c) -4.0))) b) (* a 2.0))))
   (if (<= b -8e+143)
     (/ (- b) a)
     (if (<= b 1.2e-181)
       t_0
       (if (<= b 9.5e-117)
         (*
          0.5
          (/ (- (pow (exp (* 0.25 (+ (log c) (log (* a -4.0))))) 2.0) b) a))
         (if (<= b 1.7e-26) t_0 (/ (- 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 t_0 = (sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -8e+143) {
		tmp = -b / a;
	} else if (b <= 1.2e-181) {
		tmp = t_0;
	} else if (b <= 9.5e-117) {
		tmp = 0.5 * ((pow(exp((0.25 * (log(c) + log((a * -4.0))))), 2.0) - b) / a);
	} else if (b <= 1.7e-26) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) + ((a * c) * (-4.0d0)))) - b) / (a * 2.0d0)
    if (b <= (-8d+143)) then
        tmp = -b / a
    else if (b <= 1.2d-181) then
        tmp = t_0
    else if (b <= 9.5d-117) then
        tmp = 0.5d0 * (((exp((0.25d0 * (log(c) + log((a * (-4.0d0)))))) ** 2.0d0) - b) / a)
    else if (b <= 1.7d-26) then
        tmp = t_0
    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 t_0 = (Math.sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -8e+143) {
		tmp = -b / a;
	} else if (b <= 1.2e-181) {
		tmp = t_0;
	} else if (b <= 9.5e-117) {
		tmp = 0.5 * ((Math.pow(Math.exp((0.25 * (Math.log(c) + Math.log((a * -4.0))))), 2.0) - b) / a);
	} else if (b <= 1.7e-26) {
		tmp = t_0;
	} 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):
	t_0 = (math.sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0)
	tmp = 0
	if b <= -8e+143:
		tmp = -b / a
	elif b <= 1.2e-181:
		tmp = t_0
	elif b <= 9.5e-117:
		tmp = 0.5 * ((math.pow(math.exp((0.25 * (math.log(c) + math.log((a * -4.0))))), 2.0) - b) / a)
	elif b <= 1.7e-26:
		tmp = t_0
	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)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -8e+143)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.2e-181)
		tmp = t_0;
	elseif (b <= 9.5e-117)
		tmp = Float64(0.5 * Float64(Float64((exp(Float64(0.25 * Float64(log(c) + log(Float64(a * -4.0))))) ^ 2.0) - b) / a));
	elseif (b <= 1.7e-26)
		tmp = t_0;
	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)
	t_0 = (sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
	tmp = 0.0;
	if (b <= -8e+143)
		tmp = -b / a;
	elseif (b <= 1.2e-181)
		tmp = t_0;
	elseif (b <= 9.5e-117)
		tmp = 0.5 * (((exp((0.25 * (log(c) + log((a * -4.0))))) ^ 2.0) - b) / a);
	elseif (b <= 1.7e-26)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8e+143], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.2e-181], t$95$0, If[LessEqual[b, 9.5e-117], N[(0.5 * N[(N[(N[Power[N[Exp[N[(0.25 * N[(N[Log[c], $MachinePrecision] + N[Log[N[(a * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e-26], t$95$0, N[((-c) / b), $MachinePrecision]]]]]]

Target

Original	34.3
Target	21.3
Herbie	11.0

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

Derivation

1. Split input into 4 regimes

2. if b < -8.0000000000000002e143

   1. Initial program 60.3

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

   2. Simplified 60.3

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}} \]
      Proof

      [Start] 60.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 60.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 60.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 60.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 60.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 60.3	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 60.3	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 60.3	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 60.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Taylor expanded in b around -inf 3.0

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

   4. Simplified 3.0

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

      [Start] 3.0	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 3.0	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 3.0	\[ \frac{\color{blue}{-b}}{a} \]

   if -8.0000000000000002e143 < b < 1.2000000000000001e-181 or 9.5000000000000004e-117 < b < 1.70000000000000007e-26

   1. Initial program 13.5

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

   if 1.2000000000000001e-181 < b < 9.5000000000000004e-117

   1. Initial program 22.3

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

   2. Simplified 22.3

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}} \]
      Proof

      [Start] 22.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 22.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 22.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 22.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 22.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 22.3	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 22.3	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 22.3	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 22.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Applied egg-rr 22.4

      \[\leadsto \left(\color{blue}{{\left({\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{0.25}\right)}^{2}} - b\right) \cdot \frac{0.5}{a} \]

   4. Taylor expanded in b around 0 23.1

      \[\leadsto \left({\left({\color{blue}{\left(-4 \cdot \left(c \cdot a\right)\right)}}^{0.25}\right)}^{2} - b\right) \cdot \frac{0.5}{a} \]

   5. Simplified 23.1

      \[\leadsto \left({\left({\color{blue}{\left(a \cdot \left(c \cdot -4\right)\right)}}^{0.25}\right)}^{2} - b\right) \cdot \frac{0.5}{a} \]
      Proof

      [Start] 23.1	\[ \left({\left({\left(-4 \cdot \left(c \cdot a\right)\right)}^{0.25}\right)}^{2} - b\right) \cdot \frac{0.5}{a} \]
      *-commutative [=>] 23.1	\[ \left({\left({\color{blue}{\left(\left(c \cdot a\right) \cdot -4\right)}}^{0.25}\right)}^{2} - b\right) \cdot \frac{0.5}{a} \]
      *-commutative [=>] 23.1	\[ \left({\left({\left(\color{blue}{\left(a \cdot c\right)} \cdot -4\right)}^{0.25}\right)}^{2} - b\right) \cdot \frac{0.5}{a} \]
      associate-*r* [<=] 23.1	\[ \left({\left({\color{blue}{\left(a \cdot \left(c \cdot -4\right)\right)}}^{0.25}\right)}^{2} - b\right) \cdot \frac{0.5}{a} \]

   6. Taylor expanded in c around 0 42.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{{\left(e^{0.25 \cdot \left(\log c + \log \left(-4 \cdot a\right)\right)}\right)}^{2} - b}{a}} \]

   if 1.70000000000000007e-26 < 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. Taylor expanded in b around inf 6.6

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

   3. Simplified 6.6

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

      [Start] 6.6	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 6.6	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 6.6	\[ \frac{\color{blue}{-c}}{b} \]

3. Recombined 4 regimes into one program.

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+143}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \frac{{\left(e^{0.25 \cdot \left(\log c + \log \left(a \cdot -4\right)\right)}\right)}^{2} - b}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+142}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Error	10.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+144}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{\frac{a}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Error	39.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-295}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	22.3
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-289}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Error	56.4
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023056 
(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)))