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

Average Accuracy: 77.1% → 95.3%

Time: 6.1s

Precision: binary64

Cost: 1228

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ y z) (- INFINITY))
   (/ y (/ z x))
   (if (<= (/ y z) -5e-58)
     (* (/ y z) x)
     (if (<= (/ y z) 0.0) (* y (* x (/ 1.0 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) <= -((double) INFINITY)) {
		tmp = y / (z / x);
	} else if ((y / z) <= -5e-58) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 0.0) {
		tmp = y * (x * (1.0 / z));
	} else {
		tmp = x / (z / y);
	}
	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 = y / (z / x);
	} else if ((y / z) <= -5e-58) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 0.0) {
		tmp = y * (x * (1.0 / 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) <= -math.inf:
		tmp = y / (z / x)
	elif (y / z) <= -5e-58:
		tmp = (y / z) * x
	elif (y / z) <= 0.0:
		tmp = y * (x * (1.0 / 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) <= Float64(-Inf))
		tmp = Float64(y / Float64(z / x));
	elseif (Float64(y / z) <= -5e-58)
		tmp = Float64(Float64(y / z) * x);
	elseif (Float64(y / z) <= 0.0)
		tmp = Float64(y * Float64(x * Float64(1.0 / 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) <= -Inf)
		tmp = y / (z / x);
	elseif ((y / z) <= -5e-58)
		tmp = (y / z) * x;
	elseif ((y / z) <= 0.0)
		tmp = y * (x * (1.0 / 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], (-Infinity)], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], -5e-58], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 0.0], N[(y * N[(x * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	77.1%
Target	97.5%
Herbie	95.3%

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 0.0%

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

   2. Simplified 99.6%

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

      [Start] 0.0	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      *-commutative [=>] 0.0	\[ \color{blue}{\frac{\frac{y}{z} \cdot t}{t} \cdot x} \]
      associate-/l* [=>] 0.0	\[ \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \cdot x \]
      *-inverses [=>] 0.0	\[ \frac{\frac{y}{z}}{\color{blue}{1}} \cdot x \]
      /-rgt-identity [=>] 0.0	\[ \color{blue}{\frac{y}{z}} \cdot x \]
      associate-*l/ [=>] 99.5	\[ \color{blue}{\frac{y \cdot x}{z}} \]
      associate-*r/ [<=] 99.6	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   3. Applied egg-rr 99.7%

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

      [Start] 99.6	\[ y \cdot \frac{x}{z} \]
      clear-num [=>] 99.5	\[ y \cdot \color{blue}{\frac{1}{\frac{z}{x}}} \]
      un-div-inv [=>] 99.7	\[ \color{blue}{\frac{y}{\frac{z}{x}}} \]

   if -inf.0 < (/.f64 y z) < -4.99999999999999977e-58

   1. Initial program 83.4%

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

   2. Simplified 99.6%

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

      [Start] 83.4	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 99.6	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 99.6	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 99.6	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   if -4.99999999999999977e-58 < (/.f64 y z) < -0.0

   1. Initial program 77.8%

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

   2. Simplified 95.6%

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

      [Start] 77.8	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      *-commutative [=>] 77.8	\[ \color{blue}{\frac{\frac{y}{z} \cdot t}{t} \cdot x} \]
      associate-/l* [=>] 86.6	\[ \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \cdot x \]
      *-inverses [=>] 86.6	\[ \frac{\frac{y}{z}}{\color{blue}{1}} \cdot x \]
      /-rgt-identity [=>] 86.6	\[ \color{blue}{\frac{y}{z}} \cdot x \]
      associate-*l/ [=>] 95.6	\[ \color{blue}{\frac{y \cdot x}{z}} \]
      associate-*r/ [<=] 95.6	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   3. Applied egg-rr 94.6%

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

      [Start] 95.6	\[ y \cdot \frac{x}{z} \]
      associate-*r/ [=>] 95.6	\[ \color{blue}{\frac{y \cdot x}{z}} \]
      clear-num [=>] 94.6	\[ \color{blue}{\frac{1}{\frac{z}{y \cdot x}}} \]

   4. Applied egg-rr 95.5%

      \[\leadsto \color{blue}{\left(\frac{1}{z} \cdot x\right) \cdot y} \]
      Proof

      [Start] 94.6	\[ \frac{1}{\frac{z}{y \cdot x}} \]
      associate-/r/ [=>] 95.6	\[ \color{blue}{\frac{1}{z} \cdot \left(y \cdot x\right)} \]
      *-commutative [=>] 95.6	\[ \frac{1}{z} \cdot \color{blue}{\left(x \cdot y\right)} \]
      associate-*r* [=>] 95.5	\[ \color{blue}{\left(\frac{1}{z} \cdot x\right) \cdot y} \]

   if -0.0 < (/.f64 y z)

   1. Initial program 77.0%

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

   2. Simplified 92.6%

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

      [Start] 77.0	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 92.6	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 92.6	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 92.6	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 89.1%

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

   4. Simplified 92.6%

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

      [Start] 89.1	\[ \frac{y \cdot x}{z} \]
      *-commutative [=>] 89.1	\[ \frac{\color{blue}{x \cdot y}}{z} \]
      associate-/l* [=>] 92.6	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.6%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty \lor \neg \left(\frac{y}{z} \leq -1 \cdot 10^{-213}\right) \land \frac{y}{z} \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Accuracy	96.6%
Cost	1100

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Accuracy	96.6%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Accuracy	90.3%
Cost	320

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

Reproduce

herbie shell --seed 2023151 
(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)))