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

Average Error: 23.35% → 0.85%

Time: 5.6s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-316}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y z) x)) (t_2 (* y (/ x z))))
   (if (<= (/ y z) (- INFINITY))
     t_2
     (if (<= (/ y z) -1e-147)
       t_1
       (if (<= (/ y z) 2e-316)
         t_2
         (if (<= (/ y z) 2e+185) 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 = (y / z) * x;
	double t_2 = y * (x / z);
	double tmp;
	if ((y / z) <= -((double) INFINITY)) {
		tmp = t_2;
	} else if ((y / z) <= -1e-147) {
		tmp = t_1;
	} else if ((y / z) <= 2e-316) {
		tmp = t_2;
	} else if ((y / z) <= 2e+185) {
		tmp = t_1;
	} 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 t_1 = (y / z) * x;
	double t_2 = y * (x / z);
	double tmp;
	if ((y / z) <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if ((y / z) <= -1e-147) {
		tmp = t_1;
	} else if ((y / z) <= 2e-316) {
		tmp = t_2;
	} else if ((y / z) <= 2e+185) {
		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 = (y / z) * x
	t_2 = y * (x / z)
	tmp = 0
	if (y / z) <= -math.inf:
		tmp = t_2
	elif (y / z) <= -1e-147:
		tmp = t_1
	elif (y / z) <= 2e-316:
		tmp = t_2
	elif (y / z) <= 2e+185:
		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(Float64(y / z) * x)
	t_2 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (Float64(y / z) <= Float64(-Inf))
		tmp = t_2;
	elseif (Float64(y / z) <= -1e-147)
		tmp = t_1;
	elseif (Float64(y / z) <= 2e-316)
		tmp = t_2;
	elseif (Float64(y / z) <= 2e+185)
		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 = (y / z) * x;
	t_2 = y * (x / z);
	tmp = 0.0;
	if ((y / z) <= -Inf)
		tmp = t_2;
	elseif ((y / z) <= -1e-147)
		tmp = t_1;
	elseif ((y / z) <= 2e-316)
		tmp = t_2;
	elseif ((y / z) <= 2e+185)
		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[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], (-Infinity)], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], -1e-147], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], 2e-316], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], 2e+185], t$95$1, N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]]]]

Target

Original	23.35%
Target	2.11%
Herbie	0.85%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 y z) < -inf.0 or -9.9999999999999997e-148 < (/.f64 y z) < 2.000000017e-316

   1. Initial program 33.97

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

   2. Simplified 1.52

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

      [Start] 33.97	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 25.41	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 25.41	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 25.41	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 1.39	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 1.52	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 1.52	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   if -inf.0 < (/.f64 y z) < -9.9999999999999997e-148 or 2.000000017e-316 < (/.f64 y z) < 2e185

   1. Initial program 14.46

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

   2. Simplified 0.41

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

      [Start] 14.46	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 0.41	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 0.41	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 0.41	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   if 2e185 < (/.f64 y z)

   1. Initial program 66.13

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

   2. Simplified 39.22

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

      [Start] 66.13	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 39.22	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 39.22	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 39.22	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 2.39

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

4. Final simplification 0.85

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-316}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.72%
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty \lor \neg \left(\frac{y}{z} \leq -1 \cdot 10^{-147}\right) \land \left(\frac{y}{z} \leq 2 \cdot 10^{-316} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{+255}\right)\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Error	0.71%
Cost	1360

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{-316}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 3
Error	9.85%
Cost	320

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

Reproduce

herbie shell --seed 2023115 
(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)))