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

Average Accuracy: 76.8% → 97.0%

Time: 6.6s

Precision: binary64

Cost: 1101

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-154} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{-231}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ y z) (- INFINITY))
   (/ 1.0 (/ (/ z x) y))
   (if (or (<= (/ y z) -2e-154) (not (<= (/ y z) 5e-231)))
     (/ x (/ z y))
     (/ (* 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) <= -((double) INFINITY)) {
		tmp = 1.0 / ((z / x) / y);
	} else if (((y / z) <= -2e-154) || !((y / z) <= 5e-231)) {
		tmp = x / (z / y);
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

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) <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 / ((z / x) / y);
	} else if (((y / z) <= -2e-154) || !((y / z) <= 5e-231)) {
		tmp = x / (z / y);
	} 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) <= -math.inf:
		tmp = 1.0 / ((z / x) / y)
	elif ((y / z) <= -2e-154) or not ((y / z) <= 5e-231):
		tmp = x / (z / y)
	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) <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(Float64(z / x) / y));
	elseif ((Float64(y / z) <= -2e-154) || !(Float64(y / z) <= 5e-231))
		tmp = Float64(x / Float64(z / y));
	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)
	tmp = 0.0;
	if ((y / z) <= -Inf)
		tmp = 1.0 / ((z / x) / y);
	elseif (((y / z) <= -2e-154) || ~(((y / z) <= 5e-231)))
		tmp = x / (z / y);
	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], (-Infinity)], N[(1.0 / N[(N[(z / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(y / z), $MachinePrecision], -2e-154], N[Not[LessEqual[N[(y / z), $MachinePrecision], 5e-231]], $MachinePrecision]], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	76.8%
Target	97.5%
Herbie	97.0%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 y z) < -inf.0

   1. Initial program 0.0%

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

   2. Simplified 99.7%

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

      [Start] 0.0	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 0.0	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 0.0	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 0.0	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 99.6	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 99.7	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 99.7	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{y \cdot x}}} \]
      Proof

      [Start] 99.7	\[ y \cdot \frac{x}{z} \]
      associate-*r/ [=>] 99.6	\[ \color{blue}{\frac{y \cdot x}{z}} \]
      clear-num [=>] 99.4	\[ \color{blue}{\frac{1}{\frac{z}{y \cdot x}}} \]

   4. Applied egg-rr 99.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{y} \cdot \frac{z}{x}}} \]
      Proof

      [Start] 99.4	\[ \frac{1}{\frac{z}{y \cdot x}} \]
      *-un-lft-identity [=>] 99.4	\[ \frac{1}{\frac{\color{blue}{1 \cdot z}}{y \cdot x}} \]
      times-frac [=>] 99.3	\[ \frac{1}{\color{blue}{\frac{1}{y} \cdot \frac{z}{x}}} \]

   5. Applied egg-rr 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z}{x}}{y}}} \]
      Proof

      [Start] 99.3	\[ \frac{1}{\frac{1}{y} \cdot \frac{z}{x}} \]
      associate-*l/ [=>] 99.4	\[ \frac{1}{\color{blue}{\frac{1 \cdot \frac{z}{x}}{y}}} \]
      *-un-lft-identity [<=] 99.4	\[ \frac{1}{\frac{\color{blue}{\frac{z}{x}}}{y}} \]

   if -inf.0 < (/.f64 y z) < -1.9999999999999999e-154 or 5.00000000000000023e-231 < (/.f64 y z)

   1. Initial program 81.4%

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

   2. Simplified 96.0%

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

      [Start] 81.4	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 96.0	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 96.0	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 96.0	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Applied egg-rr 96.4%

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

      [Start] 96.0	\[ x \cdot \frac{y}{z} \]
      associate-*r/ [=>] 86.6	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-/l* [=>] 96.4	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

   if -1.9999999999999999e-154 < (/.f64 y z) < 5.00000000000000023e-231

   1. Initial program 72.3%

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

   2. Simplified 83.1%

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

      [Start] 72.3	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 83.1	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 83.1	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 83.1	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 98.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-154} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{-231}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.0%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{+219} \lor \neg \left(\frac{y}{z} \leq -1 \cdot 10^{-211}\right) \land \frac{y}{z} \leq 10^{-316}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Accuracy	97.2%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty \lor \neg \left(\frac{y}{z} \leq -1 \cdot 10^{-211}\right) \land \frac{y}{z} \leq 10^{-316}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Accuracy	97.2%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-211} \lor \neg \left(\frac{y}{z} \leq 10^{-316}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 4
Accuracy	97.0%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-154} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{-231}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternative 5
Accuracy	90.1%
Cost	320

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

Reproduce

herbie shell --seed 2023138 
(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)))