bug366 (missed optimization)

Average Error: 38.1 → 12.6

Time: 1.5s

Precision: binary64

Cost: 8144

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}\\ \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-110}:\\ \;\;\;\;x \cdot -1\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x \cdot -1\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

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 (<= (* z z) 4e-110)
     (* x -1.0)
     (if (<= (* z z) 4e+28)
       t_0
       (if (<= (* z z) 2e+42) (* x -1.0) (if (<= (* z z) 5e+153) 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 ((z * z) <= 4e-110) {
		tmp = x * -1.0;
	} else if ((z * z) <= 4e+28) {
		tmp = t_0;
	} else if ((z * z) <= 2e+42) {
		tmp = x * -1.0;
	} else if ((z * z) <= 5e+153) {
		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 ((z * z) <= 4d-110) then
        tmp = x * (-1.0d0)
    else if ((z * z) <= 4d+28) then
        tmp = t_0
    else if ((z * z) <= 2d+42) then
        tmp = x * (-1.0d0)
    else if ((z * z) <= 5d+153) 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 ((z * z) <= 4e-110) {
		tmp = x * -1.0;
	} else if ((z * z) <= 4e+28) {
		tmp = t_0;
	} else if ((z * z) <= 2e+42) {
		tmp = x * -1.0;
	} else if ((z * z) <= 5e+153) {
		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 (z * z) <= 4e-110:
		tmp = x * -1.0
	elif (z * z) <= 4e+28:
		tmp = t_0
	elif (z * z) <= 2e+42:
		tmp = x * -1.0
	elif (z * z) <= 5e+153:
		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 (Float64(z * z) <= 4e-110)
		tmp = Float64(x * -1.0);
	elseif (Float64(z * z) <= 4e+28)
		tmp = t_0;
	elseif (Float64(z * z) <= 2e+42)
		tmp = Float64(x * -1.0);
	elseif (Float64(z * z) <= 5e+153)
		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 ((z * z) <= 4e-110)
		tmp = x * -1.0;
	elseif ((z * z) <= 4e+28)
		tmp = t_0;
	elseif ((z * z) <= 2e+42)
		tmp = x * -1.0;
	elseif ((z * z) <= 5e+153)
		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[N[(z * z), $MachinePrecision], 4e-110], N[(x * -1.0), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 4e+28], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 2e+42], N[(x * -1.0), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+153], t$95$0, z]]]]]

Target

Original	38.1
Target	0.0
Herbie	12.6

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.0000000000000002e-110 or 3.99999999999999983e28 < (*.f64 z z) < 2.00000000000000009e42

   1. Initial program 27.8

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

   2. Taylor expanded in x around -inf 7.8

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

   3. Simplified 7.8

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

      [Start] 7.8	\[ -1 \cdot x \]
      rational.json-simplify-2 [=>] 7.8	\[ \color{blue}{x \cdot -1} \]

   if 4.0000000000000002e-110 < (*.f64 z z) < 3.99999999999999983e28 or 2.00000000000000009e42 < (*.f64 z z) < 5.00000000000000018e153

   1. Initial program 22.6

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

   if 5.00000000000000018e153 < (*.f64 z z)

   1. Initial program 51.2

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

   2. Taylor expanded in z around inf 12.1

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

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-110}:\\ \;\;\;\;x \cdot -1\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x \cdot -1\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternatives

Alternative 1
Error	13.5
Cost	324

\[\begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{-54}:\\ \;\;\;\;x \cdot -1\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	30.8
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023053 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

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

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