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

Average Error: 22.96% → 3.99%

Time: 4.1s

Precision: binary64

Cost: 1101

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+227}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-259} \lor \neg \left(\frac{y}{z} \leq 4 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ y z) -5e+227)
   (/ (* y x) z)
   (if (or (<= (/ y z) -1e-259) (not (<= (/ y z) 4e-68)))
     (* (/ y z) x)
     (* y (/ x z)))))

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+227) {
		tmp = (y * x) / z;
	} else if (((y / z) <= -1e-259) || !((y / z) <= 4e-68)) {
		tmp = (y / z) * x;
	} else {
		tmp = y * (x / z);
	}
	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+227)) then
        tmp = (y * x) / z
    else if (((y / z) <= (-1d-259)) .or. (.not. ((y / z) <= 4d-68))) then
        tmp = (y / z) * x
    else
        tmp = y * (x / z)
    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+227) {
		tmp = (y * x) / z;
	} else if (((y / z) <= -1e-259) || !((y / z) <= 4e-68)) {
		tmp = (y / z) * x;
	} else {
		tmp = y * (x / z);
	}
	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+227:
		tmp = (y * x) / z
	elif ((y / z) <= -1e-259) or not ((y / z) <= 4e-68):
		tmp = (y / z) * x
	else:
		tmp = y * (x / z)
	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+227)
		tmp = Float64(Float64(y * x) / z);
	elseif ((Float64(y / z) <= -1e-259) || !(Float64(y / z) <= 4e-68))
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(y * Float64(x / z));
	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+227)
		tmp = (y * x) / z;
	elseif (((y / z) <= -1e-259) || ~(((y / z) <= 4e-68)))
		tmp = (y / z) * x;
	else
		tmp = y * (x / z);
	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+227], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[Or[LessEqual[N[(y / z), $MachinePrecision], -1e-259], N[Not[LessEqual[N[(y / z), $MachinePrecision], 4e-68]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	22.96%
Target	2.12%
Herbie	3.99%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 y z) < -4.9999999999999996e227

   1. Initial program 72.61

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

   2. Simplified 52.61

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

      [Start] 72.61	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 52.61	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 52.61	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 52.61	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 0.97

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

   if -4.9999999999999996e227 < (/.f64 y z) < -1.0000000000000001e-259 or 4.00000000000000027e-68 < (/.f64 y z)

   1. Initial program 18.29

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

   2. Simplified 4.99

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

      [Start] 18.29	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 4.99	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 4.99	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 4.99	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   if -1.0000000000000001e-259 < (/.f64 y z) < 4.00000000000000027e-68

   1. Initial program 24.3

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

   2. Simplified 2.6

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

      [Start] 24.3	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 13.65	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 13.65	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 13.65	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 3.6	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 2.6	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 2.6	\[ \color{blue}{y \cdot \frac{x}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 3.99

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+227}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-259} \lor \neg \left(\frac{y}{z} \leq 4 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.96%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty \lor \neg \left(\frac{y}{z} \leq -1 \cdot 10^{-259}\right) \land \frac{y}{z} \leq 4 \cdot 10^{-68}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Error	10.23%
Cost	320

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

Reproduce

herbie shell --seed 2023090 
(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)))