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

Average Error: 14.5 → 0.3

Time: 4.1s

Precision: binary64

Cost: 1361

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+284}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 4.5 \cdot 10^{-264} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{+213}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ y z) -5e+284)
   (/ y (/ z x))
   (if (<= (/ y z) -4e-208)
     (* (/ y z) x)
     (if (or (<= (/ y z) 4.5e-264) (not (<= (/ y z) 5e+213)))
       (* 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) <= -5e+284) {
		tmp = y / (z / x);
	} else if ((y / z) <= -4e-208) {
		tmp = (y / z) * x;
	} else if (((y / z) <= 4.5e-264) || !((y / z) <= 5e+213)) {
		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) <= (-5d+284)) then
        tmp = y / (z / x)
    else if ((y / z) <= (-4d-208)) then
        tmp = (y / z) * x
    else if (((y / z) <= 4.5d-264) .or. (.not. ((y / z) <= 5d+213))) 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) <= -5e+284) {
		tmp = y / (z / x);
	} else if ((y / z) <= -4e-208) {
		tmp = (y / z) * x;
	} else if (((y / z) <= 4.5e-264) || !((y / z) <= 5e+213)) {
		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) <= -5e+284:
		tmp = y / (z / x)
	elif (y / z) <= -4e-208:
		tmp = (y / z) * x
	elif ((y / z) <= 4.5e-264) or not ((y / z) <= 5e+213):
		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) <= -5e+284)
		tmp = Float64(y / Float64(z / x));
	elseif (Float64(y / z) <= -4e-208)
		tmp = Float64(Float64(y / z) * x);
	elseif ((Float64(y / z) <= 4.5e-264) || !(Float64(y / z) <= 5e+213))
		tmp = Float64(y * Float64(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) <= -5e+284)
		tmp = y / (z / x);
	elseif ((y / z) <= -4e-208)
		tmp = (y / z) * x;
	elseif (((y / z) <= 4.5e-264) || ~(((y / z) <= 5e+213)))
		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], -5e+284], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], -4e-208], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[Or[LessEqual[N[(y / z), $MachinePrecision], 4.5e-264], N[Not[LessEqual[N[(y / z), $MachinePrecision], 5e+213]], $MachinePrecision]], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	14.5
Target	1.5
Herbie	0.3

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -4.9999999999999999e284

   1. Initial program 59.2

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

   2. Taylor expanded in x around 0 0.3

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

   3. Simplified 0.2

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

      [Start] 0.3	\[ \frac{y \cdot x}{z} \]
      associate-/l* [=>] 0.2	\[ \color{blue}{\frac{y}{\frac{z}{x}}} \]

   if -4.9999999999999999e284 < (/.f64 y z) < -4.0000000000000004e-208

   1. Initial program 9.6

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

   2. Simplified 0.2

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

      [Start] 9.6	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      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 -4.0000000000000004e-208 < (/.f64 y z) < 4.5000000000000001e-264 or 4.9999999999999998e213 < (/.f64 y z)

   1. Initial program 22.4

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

   2. Simplified 0.6

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

      [Start] 22.4	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 15.2	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 15.2	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 15.2	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 0.4	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 0.6	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 0.6	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   if 4.5000000000000001e-264 < (/.f64 y z) < 4.9999999999999998e213

   1. Initial program 8.6

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

   2. Simplified 0.2

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

      [Start] 8.6	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      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	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Applied egg-rr 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+284}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 4.5 \cdot 10^{-264} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{+213}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+204} \lor \neg \left(\frac{y}{z} \leq -4 \cdot 10^{-208} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{-269}\right) \land \frac{y}{z} \leq 5 \cdot 10^{+187}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1361

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 4.5 \cdot 10^{-264} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{+213}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	6.2
Cost	320

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

Reproduce

herbie shell --seed 2023031 
(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)))