quadm (p42, negative)

Average Error: 33.8 → 10.6

Time: 14.9s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-133}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+91}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

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 -5.2e-133)
   (- (/ c b))
   (if (<= b 1.8e+91)
     (* (+ b (sqrt (- (* b b) (* c (* a 4.0))))) (/ -0.5 a))
     (+ (- (/ b 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 <= -5.2e-133) {
		tmp = -(c / b);
	} else if (b <= 1.8e+91) {
		tmp = (b + sqrt(((b * b) - (c * (a * 4.0))))) * (-0.5 / a);
	} else {
		tmp = -(b / a) + (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 <= (-5.2d-133)) then
        tmp = -(c / b)
    else if (b <= 1.8d+91) then
        tmp = (b + sqrt(((b * b) - (c * (a * 4.0d0))))) * ((-0.5d0) / a)
    else
        tmp = -(b / a) + (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 <= -5.2e-133) {
		tmp = -(c / b);
	} else if (b <= 1.8e+91) {
		tmp = (b + Math.sqrt(((b * b) - (c * (a * 4.0))))) * (-0.5 / a);
	} else {
		tmp = -(b / a) + (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 <= -5.2e-133:
		tmp = -(c / b)
	elif b <= 1.8e+91:
		tmp = (b + math.sqrt(((b * b) - (c * (a * 4.0))))) * (-0.5 / a)
	else:
		tmp = -(b / a) + (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 <= -5.2e-133)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1.8e+91)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(-Float64(b / a)) + 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 <= -5.2e-133)
		tmp = -(c / b);
	elseif (b <= 1.8e+91)
		tmp = (b + sqrt(((b * b) - (c * (a * 4.0))))) * (-0.5 / a);
	else
		tmp = -(b / a) + (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, -5.2e-133], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1.8e+91], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-N[(b / a), $MachinePrecision]) + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Target

Original	33.8
Target	20.7
Herbie	10.6

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

Derivation

1. Split input into 3 regimes

2. if b < -5.1999999999999999e-133

   1. Initial program 51.1

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

   2. Simplified 51.1

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

      [Start] 51.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 51.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 12.0

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

   4. Simplified 12.0

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

      [Start] 12.0	\[ -1 \cdot \frac{c}{b} \]
      rational.json-simplify-2 [=>] 12.0	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational.json-simplify-9 [=>] 12.0	\[ \color{blue}{-\frac{c}{b}} \]

   if -5.1999999999999999e-133 < b < 1.8e91

   1. Initial program 11.6

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

   2. Simplified 11.6

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

      [Start] 11.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 11.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 11.6

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{-\left(a + a\right)} + 0} \]

   4. Simplified 11.7

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

      [Start] 11.6	\[ \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{-\left(a + a\right)} + 0 \]
      rational.json-simplify-4 [=>] 11.6	\[ \color{blue}{\frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{-\left(a + a\right)}} \]
      rational.json-simplify-12 [=>] 11.6	\[ \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{0 - \left(a + a\right)}} \]
      rational.json-simplify-50 [=>] 11.6	\[ \color{blue}{\frac{-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\left(a + a\right) - 0}} \]
      rational.json-simplify-8 [=>] 11.6	\[ \frac{\color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{\left(a + a\right) - 0} \]
      rational.json-simplify-2 [<=] 11.6	\[ \frac{\color{blue}{-1 \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{\left(a + a\right) - 0} \]
      rational.json-simplify-5 [=>] 11.6	\[ \frac{-1 \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{a + a}} \]
      rational.json-simplify-49 [=>] 11.7	\[ \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{a + a}} \]
      rational.json-simplify-2 [=>] 11.7	\[ \left(b + \sqrt{b \cdot b - 4 \cdot \color{blue}{\left(c \cdot a\right)}}\right) \cdot \frac{-1}{a + a} \]
      rational.json-simplify-43 [=>] 11.7	\[ \left(b + \sqrt{b \cdot b - \color{blue}{c \cdot \left(a \cdot 4\right)}}\right) \cdot \frac{-1}{a + a} \]
      rational.json-simplify-35 [=>] 11.7	\[ \left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \color{blue}{\frac{-1 + -1}{\left(a + a\right) + \left(a + a\right)}} \]
      metadata-eval [=>] 11.7	\[ \left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \frac{\color{blue}{-2}}{\left(a + a\right) + \left(a + a\right)} \]

   if 1.8e91 < b

   1. Initial program 43.6

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

   2. Simplified 43.6

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

      [Start] 43.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 43.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 3.9

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

   4. Simplified 3.9

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

      [Start] 3.9	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational.json-simplify-1 [=>] 3.9	\[ \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
      rational.json-simplify-2 [=>] 3.9	\[ \color{blue}{\frac{b}{a} \cdot -1} + \frac{c}{b} \]
      rational.json-simplify-9 [=>] 3.9	\[ \color{blue}{\left(-\frac{b}{a}\right)} + \frac{c}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-133}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+91}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{a}\right) + \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-132}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;\left(\sqrt{c \cdot \left(a \cdot -4\right)} + b\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 2
Error	14.2
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-131}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-43}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 3
Error	20.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-191}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-43}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 4
Error	39.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -9200000000000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 5
Error	22.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-224}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 6
Error	57.0
Cost	192

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

Reproduce

herbie shell --seed 2023073 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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