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

Average Error: 14.7 → 0.8

Time: 3.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -2.6011659638692282 \cdot 10^{+196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -3.4793456288708597 \cdot 10^{-240}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 9.094317544538645 \cdot 10^{-189}:\\ \;\;\;\;{\left(\frac{z}{y \cdot x}\right)}^{-1}\\ \mathbf{elif}\;\frac{y}{z} \leq 2.6328892147827514 \cdot 10^{+131}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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) -2.6011659638692282e+196)
     t_1
     (if (<= (/ y z) -3.4793456288708597e-240)
       (/ x (/ z y))
       (if (<= (/ y z) 9.094317544538645e-189)
         (pow (/ z (* y x)) -1.0)
         (if (<= (/ y z) 2.6328892147827514e+131) (* (/ y z) x) 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) <= -2.6011659638692282e+196) {
		tmp = t_1;
	} else if ((y / z) <= -3.4793456288708597e-240) {
		tmp = x / (z / y);
	} else if ((y / z) <= 9.094317544538645e-189) {
		tmp = pow((z / (y * x)), -1.0);
	} else if ((y / z) <= 2.6328892147827514e+131) {
		tmp = (y / z) * x;
	} 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 * x) / z
    if ((y / z) <= (-2.6011659638692282d+196)) then
        tmp = t_1
    else if ((y / z) <= (-3.4793456288708597d-240)) then
        tmp = x / (z / y)
    else if ((y / z) <= 9.094317544538645d-189) then
        tmp = (z / (y * x)) ** (-1.0d0)
    else if ((y / z) <= 2.6328892147827514d+131) then
        tmp = (y / z) * x
    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 * x) / z;
	double tmp;
	if ((y / z) <= -2.6011659638692282e+196) {
		tmp = t_1;
	} else if ((y / z) <= -3.4793456288708597e-240) {
		tmp = x / (z / y);
	} else if ((y / z) <= 9.094317544538645e-189) {
		tmp = Math.pow((z / (y * x)), -1.0);
	} else if ((y / z) <= 2.6328892147827514e+131) {
		tmp = (y / z) * x;
	} 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) <= -2.6011659638692282e+196:
		tmp = t_1
	elif (y / z) <= -3.4793456288708597e-240:
		tmp = x / (z / y)
	elif (y / z) <= 9.094317544538645e-189:
		tmp = math.pow((z / (y * x)), -1.0)
	elif (y / z) <= 2.6328892147827514e+131:
		tmp = (y / z) * x
	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(Float64(y * x) / z)
	tmp = 0.0
	if (Float64(y / z) <= -2.6011659638692282e+196)
		tmp = t_1;
	elseif (Float64(y / z) <= -3.4793456288708597e-240)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(y / z) <= 9.094317544538645e-189)
		tmp = Float64(z / Float64(y * x)) ^ -1.0;
	elseif (Float64(y / z) <= 2.6328892147827514e+131)
		tmp = Float64(Float64(y / z) * x);
	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) <= -2.6011659638692282e+196)
		tmp = t_1;
	elseif ((y / z) <= -3.4793456288708597e-240)
		tmp = x / (z / y);
	elseif ((y / z) <= 9.094317544538645e-189)
		tmp = (z / (y * x)) ^ -1.0;
	elseif ((y / z) <= 2.6328892147827514e+131)
		tmp = (y / z) * x;
	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[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], -2.6011659638692282e+196], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], -3.4793456288708597e-240], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 9.094317544538645e-189], N[Power[N[(z / N[(y * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 2.6328892147827514e+131], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.7
Target	1.6
Herbie	0.8

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -2.6011659638692282e196 or 2.63288921478275137e131 < (/.f64 y z)

   1. Initial program 36.4

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

   2. Simplified 18.8

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

   3. Taylor expanded in x around 0 2.6

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

   if -2.6011659638692282e196 < (/.f64 y z) < -3.47934562887085969e-240

   1. Initial program 8.6

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

   2. Simplified 0.2

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

   if -3.47934562887085969e-240 < (/.f64 y z) < 9.09431754453864547e-189

   1. Initial program 17.0

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

   2. Simplified 11.9

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

   3. Applied egg-rr 1.1

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

   if 9.09431754453864547e-189 < (/.f64 y z) < 2.63288921478275137e131

   1. Initial program 7.0

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2.6011659638692282 \cdot 10^{+196}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -3.4793456288708597 \cdot 10^{-240}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 9.094317544538645 \cdot 10^{-189}:\\ \;\;\;\;{\left(\frac{z}{y \cdot x}\right)}^{-1}\\ \mathbf{elif}\;\frac{y}{z} \leq 2.6328892147827514 \cdot 10^{+131}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))