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

Average Error: 14.8 → 1.2

Time: 3.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-236}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{-z}\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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) -2e-236)
       (* (/ y z) x)
       (if (<= (/ y z) 5e-57)
         (/ (* y (- x)) (- z))
         (if (<= (/ y z) 5e+160) (/ x (/ z y)) t_1))))))

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) <= -2e-236) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 5e-57) {
		tmp = (y * -x) / -z;
	} else if ((y / z) <= 5e+160) {
		tmp = x / (z / y);
	} else {
		tmp = t_1;
	}
	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) <= -2e-236) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 5e-57) {
		tmp = (y * -x) / -z;
	} else if ((y / z) <= 5e+160) {
		tmp = x / (z / y);
	} else {
		tmp = t_1;
	}
	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) <= -2e-236:
		tmp = (y / z) * x
	elif (y / z) <= 5e-57:
		tmp = (y * -x) / -z
	elif (y / z) <= 5e+160:
		tmp = x / (z / y)
	else:
		tmp = t_1
	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(y * Float64(x / z))
	tmp = 0.0
	if (Float64(y / z) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(y / z) <= -2e-236)
		tmp = Float64(Float64(y / z) * x);
	elseif (Float64(y / z) <= 5e-57)
		tmp = Float64(Float64(y * Float64(-x)) / Float64(-z));
	elseif (Float64(y / z) <= 5e+160)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = t_1;
	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) <= -2e-236)
		tmp = (y / z) * x;
	elseif ((y / z) <= 5e-57)
		tmp = (y * -x) / -z;
	elseif ((y / z) <= 5e+160)
		tmp = x / (z / y);
	else
		tmp = t_1;
	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[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], -2e-236], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 5e-57], N[(N[(y * (-x)), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 5e+160], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.8
Target	1.6
Herbie	1.2

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -inf.0 or 5.0000000000000002e160 < (/.f64 y z)

   1. Initial program 42.7

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

   2. Simplified 26.4

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

   3. Taylor expanded in x around 0 2.0

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

   4. Simplified 1.8

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

   if -inf.0 < (/.f64 y z) < -2.0000000000000001e-236

   1. Initial program 10.9

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.2

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

   if -2.0000000000000001e-236 < (/.f64 y z) < 5.0000000000000002e-57

   1. Initial program 14.9

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

   2. Simplified 8.2

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

   3. Taylor expanded in x around 0 2.5

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

   4. Simplified 2.4

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

   5. Applied egg-rr 2.5

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

   if 5.0000000000000002e-57 < (/.f64 y z) < 5.0000000000000002e160

   1. Initial program 6.5

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-236}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{-z}\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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)))