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

Average Error: 14.5 → 1.6

Time: 5.4s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{+253}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ x z))))
   (if (<= (/ y z) -1e+101)
     t_1
     (if (<= (/ y z) -2e-54)
       (/ x (/ z y))
       (if (<= (/ y z) 0.0)
         (/ y (/ z x))
         (if (<= (/ y z) 5e+253) (* x (/ y z)) t_1))))))

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 t_1 = y * (x / z);
	double tmp;
	if ((y / z) <= -1e+101) {
		tmp = t_1;
	} else if ((y / z) <= -2e-54) {
		tmp = x / (z / y);
	} else if ((y / z) <= 0.0) {
		tmp = y / (z / x);
	} else if ((y / z) <= 5e+253) {
		tmp = x * (y / z);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = y * (x / z)
    if ((y / z) <= (-1d+101)) then
        tmp = t_1
    else if ((y / z) <= (-2d-54)) then
        tmp = x / (z / y)
    else if ((y / z) <= 0.0d0) then
        tmp = y / (z / x)
    else if ((y / z) <= 5d+253) then
        tmp = x * (y / z)
    else
        tmp = t_1
    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 t_1 = y * (x / z);
	double tmp;
	if ((y / z) <= -1e+101) {
		tmp = t_1;
	} else if ((y / z) <= -2e-54) {
		tmp = x / (z / y);
	} else if ((y / z) <= 0.0) {
		tmp = y / (z / x);
	} else if ((y / z) <= 5e+253) {
		tmp = x * (y / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = y * (x / z)
	tmp = 0
	if (y / z) <= -1e+101:
		tmp = t_1
	elif (y / z) <= -2e-54:
		tmp = x / (z / y)
	elif (y / z) <= 0.0:
		tmp = y / (z / x)
	elif (y / z) <= 5e+253:
		tmp = x * (y / z)
	else:
		tmp = t_1
	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)
	t_1 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (Float64(y / z) <= -1e+101)
		tmp = t_1;
	elseif (Float64(y / z) <= -2e-54)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(y / z) <= 0.0)
		tmp = Float64(y / Float64(z / x));
	elseif (Float64(y / z) <= 5e+253)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = t_1;
	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)
	t_1 = y * (x / z);
	tmp = 0.0;
	if ((y / z) <= -1e+101)
		tmp = t_1;
	elseif ((y / z) <= -2e-54)
		tmp = x / (z / y);
	elseif ((y / z) <= 0.0)
		tmp = y / (z / x);
	elseif ((y / z) <= 5e+253)
		tmp = x * (y / z);
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], -1e+101], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], -2e-54], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 0.0], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 5e+253], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Target

Original	14.5
Target	1.5
Herbie	1.6

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -9.9999999999999998e100 or 4.9999999999999997e253 < (/.f64 y z)

   1. Initial program 35.1

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

   2. Simplified 3.1

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

      [Start] 35.1	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 20.7	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 20.7	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 20.7	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 3.5	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 3.1	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 3.1	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   if -9.9999999999999998e100 < (/.f64 y z) < -2.0000000000000001e-54

   1. Initial program 4.2

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

   2. Simplified 0.2

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

      [Start] 4.2	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      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 0.3

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

   if -2.0000000000000001e-54 < (/.f64 y z) < 0.0

   1. Initial program 15.1

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

   2. Simplified 2.8

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

      [Start] 15.1	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 8.7	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 8.7	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 8.7	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 3.3	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 2.8	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 2.8	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   3. Applied egg-rr 2.8

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

   if 0.0 < (/.f64 y z) < 4.9999999999999997e253

   1. Initial program 9.4

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

   2. Simplified 0.5

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

      [Start] 9.4	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 0.5	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 0.5	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 0.5	\[ x \cdot \color{blue}{\frac{y}{z}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+101}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{+253}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.1
Cost	1609

\[\begin{array}{l} t_1 := x \cdot \frac{\frac{y}{z} \cdot t}{t}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq -2 \cdot 10^{-299}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+122} \lor \neg \left(\frac{y}{z} \leq -5 \cdot 10^{-163}\right) \land \left(\frac{y}{z} \leq 0 \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{+253}\right)\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 3
Error	1.4
Cost	1361

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 0 \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{+253}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 4
Error	6.2
Cost	320

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

Reproduce

herbie shell --seed 2023018 
(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)))