Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B

Average Error: 14.7 → 1.9

Time: 5.2s

Precision: binary64

Cost: 1100

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

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t} \]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+132}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-245}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-309}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (/ (* (/ y z) t) t)))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ y z) -2e+132)
   (* y (/ x z))
   (if (<= (/ y z) -2e-245)
     (* (/ y z) x)
     (if (<= (/ y z) 1e-309) (/ (* y x) z) (/ x (/ z y))))))

C

double code(double x, double y, double z, double t) {
	return x * (((y / z) * t) / t);
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y / z) <= -2e+132) {
		tmp = y * (x / z);
	} else if ((y / z) <= -2e-245) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 1e-309) {
		tmp = (y * x) / z;
	} else {
		tmp = x / (z / y);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y / z) <= (-2d+132)) then
        tmp = y * (x / z)
    else if ((y / z) <= (-2d-245)) then
        tmp = (y / z) * x
    else if ((y / z) <= 1d-309) then
        tmp = (y * x) / z
    else
        tmp = x / (z / y)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y / z) <= -2e+132) {
		tmp = y * (x / z);
	} else if ((y / z) <= -2e-245) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 1e-309) {
		tmp = (y * x) / z;
	} else {
		tmp = x / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x * (((y / z) * t) / t)

↓

def code(x, y, z, t):
	tmp = 0
	if (y / z) <= -2e+132:
		tmp = y * (x / z)
	elif (y / z) <= -2e-245:
		tmp = (y / z) * x
	elif (y / z) <= 1e-309:
		tmp = (y * x) / z
	else:
		tmp = x / (z / y)
	return tmp

Julia

function code(x, y, z, t)
	return Float64(x * Float64(Float64(Float64(y / z) * t) / t))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y / z) <= -2e+132)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(y / z) <= -2e-245)
		tmp = Float64(Float64(y / z) * x);
	elseif (Float64(y / z) <= 1e-309)
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = Float64(x / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * (((y / z) * t) / t);
end

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y / z) <= -2e+132)
		tmp = y * (x / z);
	elseif ((y / z) <= -2e-245)
		tmp = (y / z) * x;
	elseif ((y / z) <= 1e-309)
		tmp = (y * x) / z;
	else
		tmp = x / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(N[(y / z), $MachinePrecision] * t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(y / z), $MachinePrecision], -2e+132], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], -2e-245], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 1e-309], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	14.7
Target	1.7
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} < -1.20672205123045 \cdot 10^{+245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < -5.907522236933906 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 5.658954423153415 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 2.0087180502407133 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -1.99999999999999998e132

   1. Initial program 33.4

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t} \]

   2. Simplified 2.9

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]
      Proof

      [Start] 33.4	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 16.5	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 16.5	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 16.5	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 3.8	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 2.9	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 2.9	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   if -1.99999999999999998e132 < (/.f64 y z) < -1.9999999999999999e-245

   1. Initial program 7.4

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t} \]

   2. Simplified 0.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Proof

      [Start] 7.4	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 0.2	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 0.2	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 0.2	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   if -1.9999999999999999e-245 < (/.f64 y z) < 1.000000000000002e-309

   1. Initial program 17.3

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t} \]

   2. Simplified 14.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Proof

      [Start] 17.3	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 14.2	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 14.2	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 14.2	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]

   if 1.000000000000002e-309 < (/.f64 y z)

   1. Initial program 14.4

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t} \]

   2. Simplified 3.8

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Proof

      [Start] 14.4	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 3.8	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 3.8	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 3.8	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Applied egg-rr 3.5

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+132}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-245}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-309}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.2
Cost	1101

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+132} \lor \neg \left(\frac{y}{z} \leq -1 \cdot 10^{-142}\right) \land \frac{y}{z} \leq 10^{-295}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Error	2.1
Cost	1100

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-309}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	1.9
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+132}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-309}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Error	5.9
Cost	320

\[\frac{y}{z} \cdot x \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))