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

Average Error: 15.1 → 1.4

Time: 3.4s

Precision: binary64

\[[x, y] = \mathsf{sort}([x, y]) \\]

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;\frac{y}{z} \leq -3.367633030424483 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -4.160115403043379 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 5.651467744425165 \cdot 10^{-254}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 9.366731369076651 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (/ z x))))
   (if (<= (/ y z) -3.367633030424483e+163)
     t_1
     (if (<= (/ y z) -4.160115403043379e-41)
       (* (/ y z) x)
       (if (<= (/ y z) 5.651467744425165e-254)
         (* y (/ x z))
         (if (<= (/ y z) 9.366731369076651e+178) (/ 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 / (z / x);
	double tmp;
	if ((y / z) <= -3.367633030424483e+163) {
		tmp = t_1;
	} else if ((y / z) <= -4.160115403043379e-41) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 5.651467744425165e-254) {
		tmp = y * (x / z);
	} else if ((y / z) <= 9.366731369076651e+178) {
		tmp = x / (z / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * (((y / z) * t) / t)
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y / (z / x)
    if ((y / z) <= (-3.367633030424483d+163)) then
        tmp = t_1
    else if ((y / z) <= (-4.160115403043379d-41)) then
        tmp = (y / z) * x
    else if ((y / z) <= 5.651467744425165d-254) then
        tmp = y * (x / z)
    else if ((y / z) <= 9.366731369076651d+178) then
        tmp = x / (z / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

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 tmp;
	if ((y / z) <= -3.367633030424483e+163) {
		tmp = t_1;
	} else if ((y / z) <= -4.160115403043379e-41) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 5.651467744425165e-254) {
		tmp = y * (x / z);
	} else if ((y / z) <= 9.366731369076651e+178) {
		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 / (z / x)
	tmp = 0
	if (y / z) <= -3.367633030424483e+163:
		tmp = t_1
	elif (y / z) <= -4.160115403043379e-41:
		tmp = (y / z) * x
	elif (y / z) <= 5.651467744425165e-254:
		tmp = y * (x / z)
	elif (y / z) <= 9.366731369076651e+178:
		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(z / x))
	tmp = 0.0
	if (Float64(y / z) <= -3.367633030424483e+163)
		tmp = t_1;
	elseif (Float64(y / z) <= -4.160115403043379e-41)
		tmp = Float64(Float64(y / z) * x);
	elseif (Float64(y / z) <= 5.651467744425165e-254)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(y / z) <= 9.366731369076651e+178)
		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 / (z / x);
	tmp = 0.0;
	if ((y / z) <= -3.367633030424483e+163)
		tmp = t_1;
	elseif ((y / z) <= -4.160115403043379e-41)
		tmp = (y / z) * x;
	elseif ((y / z) <= 5.651467744425165e-254)
		tmp = y * (x / z);
	elseif ((y / z) <= 9.366731369076651e+178)
		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[(z / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], -3.367633030424483e+163], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], -4.160115403043379e-41], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 5.651467744425165e-254], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 9.366731369076651e+178], 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	15.1
Target	1.7
Herbie	1.4

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -3.3676330304244828e163 or 9.36673136907665148e178 < (/.f64 y z)

   1. Initial program 37.8

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

   2. Simplified 21.1

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

   3. Taylor expanded in x around 0 1.7

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

   4. Applied egg-rr 2.0

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

   5. Applied egg-rr 1.8

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

   if -3.3676330304244828e163 < (/.f64 y z) < -4.1601154030433789e-41

   1. Initial program 6.6

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

   2. Simplified 0.2

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

   if -4.1601154030433789e-41 < (/.f64 y z) < 5.6514677444251647e-254

   1. Initial program 14.7

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

   2. Simplified 8.0

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

   3. Taylor expanded in x around 0 2.9

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

   4. Applied egg-rr 2.8

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

   5. Taylor expanded in x around 0 2.8

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

   if 5.6514677444251647e-254 < (/.f64 y z) < 9.36673136907665148e178

   1. Initial program 9.0

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.2

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -3.367633030424483 \cdot 10^{+163}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq -4.160115403043379 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 5.651467744425165 \cdot 10^{-254}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 9.366731369076651 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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)))