quadp (p42, positive)

Average Error: 53.19% → 15.21%

Time: 19.2s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -7.2e+152)
   (/ (- b) a)
   (if (<= b 1.1e-100)
     (/ (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (* a 2.0))
     (/ (- 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 tmp;
	if (b <= -7.2e+152) {
		tmp = -b / a;
	} else if (b <= 1.1e-100) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} 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) :: tmp
    if (b <= (-7.2d+152)) then
        tmp = -b / a
    else if (b <= 1.1d-100) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) / (a * 2.0d0)
    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 <= -7.2e+152) {
		tmp = -b / a;
	} else if (b <= 1.1e-100) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} 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):
	tmp = 0
	if b <= -7.2e+152:
		tmp = -b / a
	elif b <= 1.1e-100:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0)
	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)
	tmp = 0.0
	if (b <= -7.2e+152)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.1e-100)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) / Float64(a * 2.0));
	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)
	tmp = 0.0;
	if (b <= -7.2e+152)
		tmp = -b / a;
	elseif (b <= 1.1e-100)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	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_] := If[LessEqual[b, -7.2e+152], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.1e-100], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Target

Original	53.19%
Target	32.4%
Herbie	15.21%

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

Derivation

1. Split input into 3 regimes

2. if b < -7.1999999999999998e152

   1. Initial program 98.8

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

   2. Simplified 98.8

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

      [Start] 98.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 98.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 98.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 98.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 98.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 98.8	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 98.8	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 98.8	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 98.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Taylor expanded in b around -inf 3.36

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

   4. Simplified 3.36

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

      [Start] 3.36	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 3.36	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 3.36	\[ \frac{\color{blue}{-b}}{a} \]

   if -7.1999999999999998e152 < b < 1.09999999999999995e-100

   1. Initial program 17.42

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

   2. Simplified 17.4

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

      [Start] 17.42	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      +-commutative [=>] 17.42	\[ \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]
      unsub-neg [=>] 17.42	\[ \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a} \]
      fma-neg [=>] 17.42	\[ \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} - b}{2 \cdot a} \]
      *-commutative [=>] 17.42	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b}{2 \cdot a} \]
      distribute-rgt-neg-in [=>] 17.42	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{2 \cdot a} \]
      associate-*l* [=>] 17.4	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)} - b}{2 \cdot a} \]
      metadata-eval [=>] 17.4	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)} - b}{2 \cdot a} \]
      *-commutative [=>] 17.4	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 17.4

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

   if 1.09999999999999995e-100 < b

   1. Initial program 81.04

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

   2. Taylor expanded in b around inf 16

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

   3. Simplified 16

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

      [Start] 16	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 16	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 16	\[ \frac{\color{blue}{-c}}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 15.21

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.56%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-100}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Error	21.64%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-13}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-119}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Error	21.57%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Error	34.39%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Error	61%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	34.39%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-304}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Error	97.42%
Cost	192

\[\frac{b}{a} \]

Alternative 8
Error	88.29%
Cost	192

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

Reproduce

herbie shell --seed 2023089 
(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)))