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

Average Error: 22.54% → 3.29%

Time: 5.2s

Precision: binary64

Cost: 1100

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-232}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y x) z)))
   (if (<= (/ y z) (- INFINITY))
     t_1
     (if (<= (/ y z) -1e-232)
       (* (/ y z) x)
       (if (<= (/ y z) 0.0) t_1 (/ 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 t_1 = (y * x) / z;
	double tmp;
	if ((y / z) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((y / z) <= -1e-232) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 0.0) {
		tmp = t_1;
	} 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 t_1 = (y * x) / z;
	double tmp;
	if ((y / z) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((y / z) <= -1e-232) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 0.0) {
		tmp = t_1;
	} 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):
	t_1 = (y * x) / z
	tmp = 0
	if (y / z) <= -math.inf:
		tmp = t_1
	elif (y / z) <= -1e-232:
		tmp = (y / z) * x
	elif (y / z) <= 0.0:
		tmp = t_1
	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)
	t_1 = Float64(Float64(y * x) / z)
	tmp = 0.0
	if (Float64(y / z) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(y / z) <= -1e-232)
		tmp = Float64(Float64(y / z) * x);
	elseif (Float64(y / z) <= 0.0)
		tmp = t_1;
	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)
	t_1 = (y * x) / z;
	tmp = 0.0;
	if ((y / z) <= -Inf)
		tmp = t_1;
	elseif ((y / z) <= -1e-232)
		tmp = (y / z) * x;
	elseif ((y / z) <= 0.0)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], -1e-232], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 0.0], t$95$1, N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	22.54%
Target	2.37%
Herbie	3.29%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 y z) < -inf.0 or -1.00000000000000002e-232 < (/.f64 y z) < 0.0

   1. Initial program 37.13

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

   2. Simplified 32.52

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

      [Start] 37.13	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 32.52	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 32.52	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 32.52	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 0.31

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

   if -inf.0 < (/.f64 y z) < -1.00000000000000002e-232

   1. Initial program 15.55

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

   2. Simplified 0.37

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

      [Start] 15.55	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 0.37	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 0.37	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 0.37	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   if 0.0 < (/.f64 y z)

   1. Initial program 21.28

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

   2. Simplified 7.07

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

      [Start] 21.28	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 7.07	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 7.07	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 7.07	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Applied egg-rr 7.08

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

4. Final simplification 3.29

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-232}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 0:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.31%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+281} \lor \neg \left(\frac{y}{z} \leq -5 \cdot 10^{-248}\right) \land \frac{y}{z} \leq 10^{-142}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Error	3.06%
Cost	1100

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -5 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 4 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	3.06%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq -5 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 4 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Error	9.97%
Cost	320

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

Reproduce

herbie shell --seed 2023121 
(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)))