bug366 (missed optimization)

Average Error: 38.3 → 11.3

Time: 2.6s

Precision: binary64

Cost: 7632

\[ \begin{array}{c}[x, y, z] = \mathsf{sort}([x, y, z])\\ \end{array} \]

Math

\[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

↓

\[\begin{array}{l} t_0 := \sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+146}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -132000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (sqrt (+ (* x x) (+ (* y y) (* z z)))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* x x) (+ (* y y) (* z z))))))
   (if (<= x -7.2e+146)
     (- x)
     (if (<= x -132000000.0)
       t_0
       (if (<= x -8.6e-8) z (if (<= x -3.8e-65) t_0 z))))))

C

double code(double x, double y, double z) {
	return sqrt(((x * x) + ((y * y) + (z * z))));
}

↓

double code(double x, double y, double z) {
	double t_0 = sqrt(((x * x) + ((y * y) + (z * z))));
	double tmp;
	if (x <= -7.2e+146) {
		tmp = -x;
	} else if (x <= -132000000.0) {
		tmp = t_0;
	} else if (x <= -8.6e-8) {
		tmp = z;
	} else if (x <= -3.8e-65) {
		tmp = t_0;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt(((x * x) + ((y * y) + (z * z))))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((x * x) + ((y * y) + (z * z))))
    if (x <= (-7.2d+146)) then
        tmp = -x
    else if (x <= (-132000000.0d0)) then
        tmp = t_0
    else if (x <= (-8.6d-8)) then
        tmp = z
    else if (x <= (-3.8d-65)) then
        tmp = t_0
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return Math.sqrt(((x * x) + ((y * y) + (z * z))));
}

↓

public static double code(double x, double y, double z) {
	double t_0 = Math.sqrt(((x * x) + ((y * y) + (z * z))));
	double tmp;
	if (x <= -7.2e+146) {
		tmp = -x;
	} else if (x <= -132000000.0) {
		tmp = t_0;
	} else if (x <= -8.6e-8) {
		tmp = z;
	} else if (x <= -3.8e-65) {
		tmp = t_0;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	return math.sqrt(((x * x) + ((y * y) + (z * z))))

↓

def code(x, y, z):
	t_0 = math.sqrt(((x * x) + ((y * y) + (z * z))))
	tmp = 0
	if x <= -7.2e+146:
		tmp = -x
	elif x <= -132000000.0:
		tmp = t_0
	elif x <= -8.6e-8:
		tmp = z
	elif x <= -3.8e-65:
		tmp = t_0
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	return sqrt(Float64(Float64(x * x) + Float64(Float64(y * y) + Float64(z * z))))
end

↓

function code(x, y, z)
	t_0 = sqrt(Float64(Float64(x * x) + Float64(Float64(y * y) + Float64(z * z))))
	tmp = 0.0
	if (x <= -7.2e+146)
		tmp = Float64(-x);
	elseif (x <= -132000000.0)
		tmp = t_0;
	elseif (x <= -8.6e-8)
		tmp = z;
	elseif (x <= -3.8e-65)
		tmp = t_0;
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = sqrt(((x * x) + ((y * y) + (z * z))));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = sqrt(((x * x) + ((y * y) + (z * z))));
	tmp = 0.0;
	if (x <= -7.2e+146)
		tmp = -x;
	elseif (x <= -132000000.0)
		tmp = t_0;
	elseif (x <= -8.6e-8)
		tmp = z;
	elseif (x <= -3.8e-65)
		tmp = t_0;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -7.2e+146], (-x), If[LessEqual[x, -132000000.0], t$95$0, If[LessEqual[x, -8.6e-8], z, If[LessEqual[x, -3.8e-65], t$95$0, z]]]]]

Target

Original	38.3
Target	0.0
Herbie	11.3

\[\mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right) \]

Derivation

1. Split input into 3 regimes

2. if x < -7.1999999999999997e146

   1. Initial program 62.0

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

   2. Simplified 62.0

      \[\leadsto \color{blue}{\sqrt{z \cdot z + \left(x \cdot x + y \cdot y\right)}} \]
      Proof

      [Start] 62.0	\[ \sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
      rational.json-simplify-41 [<=] 62.0	\[ \sqrt{\color{blue}{z \cdot z + \left(x \cdot x + y \cdot y\right)}} \]

   3. Taylor expanded in x around -inf 8.0

      \[\leadsto \color{blue}{-1 \cdot x} \]

   4. Simplified 8.0

      \[\leadsto \color{blue}{-x} \]
      Proof

      [Start] 8.0	\[ -1 \cdot x \]
      rational.json-simplify-2 [=>] 8.0	\[ \color{blue}{x \cdot -1} \]
      rational.json-simplify-9 [=>] 8.0	\[ \color{blue}{-x} \]

   if -7.1999999999999997e146 < x < -1.32e8 or -8.6000000000000002e-8 < x < -3.8000000000000002e-65

   1. Initial program 19.9

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

   if -1.32e8 < x < -8.6000000000000002e-8 or -3.8000000000000002e-65 < x

   1. Initial program 31.1

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

   2. Simplified 31.1

      \[\leadsto \color{blue}{\sqrt{z \cdot z + \left(x \cdot x + y \cdot y\right)}} \]
      Proof

      [Start] 31.1	\[ \sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
      rational.json-simplify-41 [<=] 31.1	\[ \sqrt{\color{blue}{z \cdot z + \left(x \cdot x + y \cdot y\right)}} \]

   3. Taylor expanded in z around inf 7.1

      \[\leadsto \color{blue}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+146}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -132000000:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternatives

Alternative 1
Error	12.9
Cost	260

\[\begin{array}{l} \mathbf{if}\;x \leq -15500000000:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	31.0
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

  :herbie-target
  (hypot x (hypot y z))

  (sqrt (+ (* x x) (+ (* y y) (* z z)))))