AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1

Average Error: 26.9 → 1.6

Time: 8.9s

Precision: binary64

Math

\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

↓

\[\begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{a}{t_1}\\ t_3 := \frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{t_1}\\ t_4 := \frac{y}{\frac{t_1}{z}} + \left(\left(x \cdot \frac{z}{t_1} + \left(t \cdot t_2 + y \cdot t_2\right)\right) - b \cdot \frac{y}{t_1}\right)\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{+141}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+242}:\\ \;\;\;\;\left(\frac{y \cdot z}{t_1} + \left(\frac{x \cdot z}{t_1} + \left(\frac{t \cdot a}{t_1} + \frac{y \cdot a}{t_1}\right)\right)\right) - \frac{y \cdot b}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t)))
        (t_2 (/ a t_1))
        (t_3 (/ (- (+ (* (+ x y) z) (* (+ y t) a)) (* y b)) t_1))
        (t_4
         (+
          (/ y (/ t_1 z))
          (- (+ (* x (/ z t_1)) (+ (* t t_2) (* y t_2))) (* b (/ y t_1))))))
   (if (<= t_3 -2e+141)
     t_4
     (if (<= t_3 5e+242)
       (-
        (+
         (/ (* y z) t_1)
         (+ (/ (* x z) t_1) (+ (/ (* t a) t_1) (/ (* y a) t_1))))
        (/ (* y b) t_1))
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = a / t_1;
	double t_3 = ((((x + y) * z) + ((y + t) * a)) - (y * b)) / t_1;
	double t_4 = (y / (t_1 / z)) + (((x * (z / t_1)) + ((t * t_2) + (y * t_2))) - (b * (y / t_1)));
	double tmp;
	if (t_3 <= -2e+141) {
		tmp = t_4;
	} else if (t_3 <= 5e+242) {
		tmp = (((y * z) / t_1) + (((x * z) / t_1) + (((t * a) / t_1) + ((y * a) / t_1)))) - ((y * b) / t_1);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
end function

↓

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = y + (x + t)
    t_2 = a / t_1
    t_3 = ((((x + y) * z) + ((y + t) * a)) - (y * b)) / t_1
    t_4 = (y / (t_1 / z)) + (((x * (z / t_1)) + ((t * t_2) + (y * t_2))) - (b * (y / t_1)))
    if (t_3 <= (-2d+141)) then
        tmp = t_4
    else if (t_3 <= 5d+242) then
        tmp = (((y * z) / t_1) + (((x * z) / t_1) + (((t * a) / t_1) + ((y * a) / t_1)))) - ((y * b) / t_1)
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = a / t_1;
	double t_3 = ((((x + y) * z) + ((y + t) * a)) - (y * b)) / t_1;
	double t_4 = (y / (t_1 / z)) + (((x * (z / t_1)) + ((t * t_2) + (y * t_2))) - (b * (y / t_1)));
	double tmp;
	if (t_3 <= -2e+141) {
		tmp = t_4;
	} else if (t_3 <= 5e+242) {
		tmp = (((y * z) / t_1) + (((x * z) / t_1) + (((t * a) / t_1) + ((y * a) / t_1)))) - ((y * b) / t_1);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)

↓

def code(x, y, z, t, a, b):
	t_1 = y + (x + t)
	t_2 = a / t_1
	t_3 = ((((x + y) * z) + ((y + t) * a)) - (y * b)) / t_1
	t_4 = (y / (t_1 / z)) + (((x * (z / t_1)) + ((t * t_2) + (y * t_2))) - (b * (y / t_1)))
	tmp = 0
	if t_3 <= -2e+141:
		tmp = t_4
	elif t_3 <= 5e+242:
		tmp = (((y * z) / t_1) + (((x * z) / t_1) + (((t * a) / t_1) + ((y * a) / t_1)))) - ((y * b) / t_1)
	else:
		tmp = t_4
	return tmp

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
end

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	t_2 = Float64(a / t_1)
	t_3 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(y + t) * a)) - Float64(y * b)) / t_1)
	t_4 = Float64(Float64(y / Float64(t_1 / z)) + Float64(Float64(Float64(x * Float64(z / t_1)) + Float64(Float64(t * t_2) + Float64(y * t_2))) - Float64(b * Float64(y / t_1))))
	tmp = 0.0
	if (t_3 <= -2e+141)
		tmp = t_4;
	elseif (t_3 <= 5e+242)
		tmp = Float64(Float64(Float64(Float64(y * z) / t_1) + Float64(Float64(Float64(x * z) / t_1) + Float64(Float64(Float64(t * a) / t_1) + Float64(Float64(y * a) / t_1)))) - Float64(Float64(y * b) / t_1));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
end

↓

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (x + t);
	t_2 = a / t_1;
	t_3 = ((((x + y) * z) + ((y + t) * a)) - (y * b)) / t_1;
	t_4 = (y / (t_1 / z)) + (((x * (z / t_1)) + ((t * t_2) + (y * t_2))) - (b * (y / t_1)));
	tmp = 0.0;
	if (t_3 <= -2e+141)
		tmp = t_4;
	elseif (t_3 <= 5e+242)
		tmp = (((y * z) / t_1) + (((x * z) / t_1) + (((t * a) / t_1) + ((y * a) / t_1)))) - ((y * b) / t_1);
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(y + t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(t * t$95$2), $MachinePrecision] + N[(y * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+141], t$95$4, If[LessEqual[t$95$3, 5e+242], N[(N[(N[(N[(y * z), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(N[(x * z), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(N[(t * a), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(y * a), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	26.9
Target	10.8
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2.00000000000000003e141 or 5.0000000000000004e242 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 53.5

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Simplified 53.5

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, \left(z + a\right) - b, x \cdot z\right)\right)}{x + \left(y + t\right)}} \]

   3. Taylor expanded in a around 0 53.5

      \[\leadsto \color{blue}{\left(\frac{y \cdot z}{y + \left(t + x\right)} + \left(\frac{z \cdot x}{y + \left(t + x\right)} + \left(\frac{a \cdot t}{y + \left(t + x\right)} + \frac{y \cdot a}{y + \left(t + x\right)}\right)\right)\right) - \frac{y \cdot b}{y + \left(t + x\right)}} \]

   4. Simplified 2.9

      \[\leadsto \color{blue}{\frac{y}{\frac{y + \left(x + t\right)}{z}} + \left(\left(\frac{z}{y + \left(x + t\right)} \cdot x + \left(\frac{a}{y + \left(x + t\right)} \cdot t + \frac{a}{y + \left(x + t\right)} \cdot y\right)\right) - \frac{y}{y + \left(x + t\right)} \cdot b\right)} \]

   if -2.00000000000000003e141 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000004e242

   1. Initial program 0.3

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, \left(z + a\right) - b, x \cdot z\right)\right)}{x + \left(y + t\right)}} \]

   3. Taylor expanded in a around 0 0.3

      \[\leadsto \color{blue}{\left(\frac{y \cdot z}{y + \left(t + x\right)} + \left(\frac{z \cdot x}{y + \left(t + x\right)} + \left(\frac{a \cdot t}{y + \left(t + x\right)} + \frac{y \cdot a}{y + \left(t + x\right)}\right)\right)\right) - \frac{y \cdot b}{y + \left(t + x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\frac{y}{\frac{y + \left(x + t\right)}{z}} + \left(\left(x \cdot \frac{z}{y + \left(x + t\right)} + \left(t \cdot \frac{a}{y + \left(x + t\right)} + y \cdot \frac{a}{y + \left(x + t\right)}\right)\right) - b \cdot \frac{y}{y + \left(x + t\right)}\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 5 \cdot 10^{+242}:\\ \;\;\;\;\left(\frac{y \cdot z}{y + \left(x + t\right)} + \left(\frac{x \cdot z}{y + \left(x + t\right)} + \left(\frac{t \cdot a}{y + \left(x + t\right)} + \frac{y \cdot a}{y + \left(x + t\right)}\right)\right)\right) - \frac{y \cdot b}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y + \left(x + t\right)}{z}} + \left(\left(x \cdot \frac{z}{y + \left(x + t\right)} + \left(t \cdot \frac{a}{y + \left(x + t\right)} + y \cdot \frac{a}{y + \left(x + t\right)}\right)\right) - b \cdot \frac{y}{y + \left(x + t\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1.0 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b)))

  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))