quadp (p42, positive)

Average Accuracy: 46.1% → 84.5%

Time: 20.4s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -2.2e+153)
   (- (/ c b) (/ b a))
   (if (<= b 2.7e-68)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) 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 <= -2.2e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.7e-68) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - 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 <= (-2.2d+153)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.7d-68) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - 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 <= -2.2e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.7e-68) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - 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 <= -2.2e+153:
		tmp = (c / b) - (b / a)
	elif b <= 2.7e-68:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - 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 <= -2.2e+153)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.7e-68)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - 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 <= -2.2e+153)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.7e-68)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - 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, -2.2e+153], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e-68], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Target

Original	46.1%
Target	67.7%
Herbie	84.5%

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

Derivation

1. Split input into 3 regimes

2. if b < -2.2e153

   1. Initial program 0.5%

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

   2. Simplified 0.5%

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

      [Start] 0.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 0.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 0.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [<=] 0.5	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 0.5	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      +-commutative [=>] 0.5	\[ \color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]
      unsub-neg [=>] 0.5	\[ \color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)} \cdot \frac{--1}{2 \cdot a} \]
      fma-neg [=>] 0.5	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      *-commutative [=>] 0.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      distribute-rgt-neg-in [=>] 0.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      associate-*l* [=>] 0.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      metadata-eval [=>] 0.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      associate-/r* [=>] 0.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]
      metadata-eval [=>] 0.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
      metadata-eval [=>] 0.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Taylor expanded in b around -inf 96.4%

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

   4. Simplified 96.4%

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

      [Start] 96.4	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 96.4	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 96.4	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -2.2e153 < b < 2.7000000000000002e-68

   1. Initial program 80.2%

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

   if 2.7000000000000002e-68 < b

   1. Initial program 16.6%

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

   2. Simplified 16.6%

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

      [Start] 16.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 16.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 16.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [<=] 16.6	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 16.6	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2 \cdot a}} \]
      +-commutative [=>] 16.6	\[ \color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]
      unsub-neg [=>] 16.6	\[ \color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)} \cdot \frac{--1}{2 \cdot a} \]
      fma-neg [=>] 16.6	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      *-commutative [=>] 16.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      distribute-rgt-neg-in [=>] 16.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      associate-*l* [=>] 16.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      metadata-eval [=>] 16.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      associate-/r* [=>] 16.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]
      metadata-eval [=>] 16.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
      metadata-eval [=>] 16.6	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Taylor expanded in b around inf 86.6%

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

   4. Simplified 86.6%

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

      [Start] 86.6	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 86.6	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      mul-1-neg [=>] 86.6	\[ \frac{\color{blue}{-c}}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.0%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+70}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-68}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Accuracy	79.1%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-19}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(c \cdot a\right) \cdot -4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Accuracy	79.2%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -9.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Accuracy	37.5%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 5
Accuracy	64.8%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-169}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Accuracy	11.6%
Cost	192

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

Reproduce

herbie shell --seed 2023146 
(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)))