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

Average Error: 14.4 → 0.4

Time: 4.9s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+295}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (/ z y))))
   (if (<= (/ y z) -2e+262)
     (/ y (/ z x))
     (if (<= (/ y z) -1e-165)
       t_1
       (if (<= (/ y z) 5e-285)
         (* y (/ x z))
         (if (<= (/ y z) 1e+295) t_1 (/ (* 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 t_1 = x / (z / y);
	double tmp;
	if ((y / z) <= -2e+262) {
		tmp = y / (z / x);
	} else if ((y / z) <= -1e-165) {
		tmp = t_1;
	} else if ((y / z) <= 5e-285) {
		tmp = y * (x / z);
	} else if ((y / z) <= 1e+295) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = x / (z / y)
    if ((y / z) <= (-2d+262)) then
        tmp = y / (z / x)
    else if ((y / z) <= (-1d-165)) then
        tmp = t_1
    else if ((y / z) <= 5d-285) then
        tmp = y * (x / z)
    else if ((y / z) <= 1d+295) then
        tmp = t_1
    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 t_1 = x / (z / y);
	double tmp;
	if ((y / z) <= -2e+262) {
		tmp = y / (z / x);
	} else if ((y / z) <= -1e-165) {
		tmp = t_1;
	} else if ((y / z) <= 5e-285) {
		tmp = y * (x / z);
	} else if ((y / z) <= 1e+295) {
		tmp = t_1;
	} 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):
	t_1 = x / (z / y)
	tmp = 0
	if (y / z) <= -2e+262:
		tmp = y / (z / x)
	elif (y / z) <= -1e-165:
		tmp = t_1
	elif (y / z) <= 5e-285:
		tmp = y * (x / z)
	elif (y / z) <= 1e+295:
		tmp = t_1
	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)
	t_1 = Float64(x / Float64(z / y))
	tmp = 0.0
	if (Float64(y / z) <= -2e+262)
		tmp = Float64(y / Float64(z / x));
	elseif (Float64(y / z) <= -1e-165)
		tmp = t_1;
	elseif (Float64(y / z) <= 5e-285)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(y / z) <= 1e+295)
		tmp = t_1;
	else
		tmp = Float64(Float64(y * 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)
	t_1 = x / (z / y);
	tmp = 0.0;
	if ((y / z) <= -2e+262)
		tmp = y / (z / x);
	elseif ((y / z) <= -1e-165)
		tmp = t_1;
	elseif ((y / z) <= 5e-285)
		tmp = y * (x / z);
	elseif ((y / z) <= 1e+295)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], -2e+262], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], -1e-165], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], 5e-285], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 1e+295], t$95$1, N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]]]

Target

Original	14.4
Target	1.6
Herbie	0.4

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -2e262

   1. Initial program 53.7

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

   2. Simplified 0.2

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

      [Start] 53.7	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 42.5	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 42.5	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 42.5	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 0.5	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 0.2	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 0.2	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   3. Applied egg-rr 0.3

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

   if -2e262 < (/.f64 y z) < -1e-165 or 5.00000000000000018e-285 < (/.f64 y z) < 9.9999999999999998e294

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

   3. Applied egg-rr 0.2

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

   if -1e-165 < (/.f64 y z) < 5.00000000000000018e-285

   1. Initial program 16.5

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

   2. Simplified 0.8

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

      [Start] 16.5	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 11.4	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 11.4	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 11.4	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 0.7	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 0.8	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 0.8	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   if 9.9999999999999998e294 < (/.f64 y z)

   1. Initial program 58.3

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

   2. Simplified 55.3

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

      [Start] 58.3	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 55.3	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 55.3	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 55.3	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-165}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{+295}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty \lor \neg \left(\frac{y}{z} \leq -2 \cdot 10^{-124} \lor \neg \left(\frac{y}{z} \leq 2 \cdot 10^{-291}\right) \land \frac{y}{z} \leq 2 \cdot 10^{+237}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+278} \lor \neg \left(\frac{y}{z} \leq -1 \cdot 10^{-165}\right) \land \left(\frac{y}{z} \leq 5 \cdot 10^{-285} \lor \neg \left(\frac{y}{z} \leq 2 \cdot 10^{+232}\right)\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	1360

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y}}\\ t_2 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+232}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	6.3
Cost	320

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

Reproduce

herbie shell --seed 2023016 
(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)))