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

Average Error: 15.2 → 0.4

Time: 4.2s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y}}\\ t_2 := \frac{y \cdot x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -5 \cdot 10^{-209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-319}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+226}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))) (t_2 (/ (* y x) z)))
   (if (<= (/ y z) (- INFINITY))
     (* y (/ x z))
     (if (<= (/ y z) -5e-209)
       t_1
       (if (<= (/ y z) 1e-319) t_2 (if (<= (/ y z) 2e+226) t_1 t_2))))))

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 t_2 = (y * x) / z;
	double tmp;
	if ((y / z) <= -((double) INFINITY)) {
		tmp = y * (x / z);
	} else if ((y / z) <= -5e-209) {
		tmp = t_1;
	} else if ((y / z) <= 1e-319) {
		tmp = t_2;
	} else if ((y / z) <= 2e+226) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 = x / (z / y);
	double t_2 = (y * x) / z;
	double tmp;
	if ((y / z) <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x / z);
	} else if ((y / z) <= -5e-209) {
		tmp = t_1;
	} else if ((y / z) <= 1e-319) {
		tmp = t_2;
	} else if ((y / z) <= 2e+226) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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)
	t_2 = (y * x) / z
	tmp = 0
	if (y / z) <= -math.inf:
		tmp = y * (x / z)
	elif (y / z) <= -5e-209:
		tmp = t_1
	elif (y / z) <= 1e-319:
		tmp = t_2
	elif (y / z) <= 2e+226:
		tmp = t_1
	else:
		tmp = t_2
	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))
	t_2 = Float64(Float64(y * x) / z)
	tmp = 0.0
	if (Float64(y / z) <= Float64(-Inf))
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(y / z) <= -5e-209)
		tmp = t_1;
	elseif (Float64(y / z) <= 1e-319)
		tmp = t_2;
	elseif (Float64(y / z) <= 2e+226)
		tmp = t_1;
	else
		tmp = t_2;
	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);
	t_2 = (y * x) / z;
	tmp = 0.0;
	if ((y / z) <= -Inf)
		tmp = y * (x / z);
	elseif ((y / z) <= -5e-209)
		tmp = t_1;
	elseif ((y / z) <= 1e-319)
		tmp = t_2;
	elseif ((y / z) <= 2e+226)
		tmp = t_1;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], (-Infinity)], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], -5e-209], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], 1e-319], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], 2e+226], t$95$1, t$95$2]]]]]]

Target

Original	15.2
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 3 regimes

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

   1. Initial program 64.0

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

   2. Taylor expanded in x around 0 0.2

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

   3. Simplified 0.3

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

   if -inf.0 < (/.f64 y z) < -5.0000000000000005e-209 or 9.99989e-320 < (/.f64 y z) < 1.99999999999999992e226

   1. Initial program 9.9

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

   2. Simplified 0.4

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

   if -5.0000000000000005e-209 < (/.f64 y z) < 9.99989e-320 or 1.99999999999999992e226 < (/.f64 y z)

   1. Initial program 25.1

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

   2. Simplified 17.8

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

   3. Taylor expanded in x around 0 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -5 \cdot 10^{-209}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-319}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+226}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+262}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternative 2
Error	6.0
Cost	320

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

Reproduce

herbie shell --seed 2022228 
(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)))