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

Average Error: 14.7 → 0.6

Time: 11.9s

Precision: binary64

Cost: 1488

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := \frac{1}{\frac{z}{y \cdot x}}\\ \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+256}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-253}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y z) x)) (t_2 (/ 1.0 (/ z (* y x)))))
   (if (<= (/ y z) -1e+256)
     t_2
     (if (<= (/ y z) -1e-242)
       t_1
       (if (<= (/ y z) 1e-253)
         (/ (* y x) z)
         (if (<= (/ y z) 5e+109) 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 = (y / z) * x;
	double t_2 = 1.0 / (z / (y * x));
	double tmp;
	if ((y / z) <= -1e+256) {
		tmp = t_2;
	} else if ((y / z) <= -1e-242) {
		tmp = t_1;
	} else if ((y / z) <= 1e-253) {
		tmp = (y * x) / z;
	} else if ((y / z) <= 5e+109) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (y / z) * x
    t_2 = 1.0d0 / (z / (y * x))
    if ((y / z) <= (-1d+256)) then
        tmp = t_2
    else if ((y / z) <= (-1d-242)) then
        tmp = t_1
    else if ((y / z) <= 1d-253) then
        tmp = (y * x) / z
    else if ((y / z) <= 5d+109) then
        tmp = t_1
    else
        tmp = t_2
    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 t_2 = 1.0 / (z / (y * x));
	double tmp;
	if ((y / z) <= -1e+256) {
		tmp = t_2;
	} else if ((y / z) <= -1e-242) {
		tmp = t_1;
	} else if ((y / z) <= 1e-253) {
		tmp = (y * x) / z;
	} else if ((y / z) <= 5e+109) {
		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 = (y / z) * x
	t_2 = 1.0 / (z / (y * x))
	tmp = 0
	if (y / z) <= -1e+256:
		tmp = t_2
	elif (y / z) <= -1e-242:
		tmp = t_1
	elif (y / z) <= 1e-253:
		tmp = (y * x) / z
	elif (y / z) <= 5e+109:
		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(Float64(y / z) * x)
	t_2 = Float64(1.0 / Float64(z / Float64(y * x)))
	tmp = 0.0
	if (Float64(y / z) <= -1e+256)
		tmp = t_2;
	elseif (Float64(y / z) <= -1e-242)
		tmp = t_1;
	elseif (Float64(y / z) <= 1e-253)
		tmp = Float64(Float64(y * x) / z);
	elseif (Float64(y / z) <= 5e+109)
		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 = (y / z) * x;
	t_2 = 1.0 / (z / (y * x));
	tmp = 0.0;
	if ((y / z) <= -1e+256)
		tmp = t_2;
	elseif ((y / z) <= -1e-242)
		tmp = t_1;
	elseif ((y / z) <= 1e-253)
		tmp = (y * x) / z;
	elseif ((y / z) <= 5e+109)
		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[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(z / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], -1e+256], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], -1e-242], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], 1e-253], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 5e+109], t$95$1, t$95$2]]]]]]

Target

Original	14.7
Target	1.4
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 y z) < -1e256 or 5.0000000000000001e109 < (/.f64 y z)

   1. Initial program 34.6

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

   2. Simplified 2.8

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]
      Proof
      (*.f64 y (/.f64 x z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y x) z)): 52 points increase in error, 62 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 y z) x)): 52 points increase in error, 55 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 x (/.f64 y z))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= /-rgt-identity_binary64 (/.f64 (/.f64 y z) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 y z) (Rewrite<= *-inverses_binary64 (/.f64 t t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 y z) t) t))): 51 points increase in error, 0 points decrease in error

   3. Applied egg-rr 3.2

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

   if -1e256 < (/.f64 y z) < -1e-242 or 1.0000000000000001e-253 < (/.f64 y z) < 5.0000000000000001e109

   1. Initial program 8.8

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Proof
      (*.f64 x (/.f64 y z)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= /-rgt-identity_binary64 (/.f64 (/.f64 y z) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 y z) (Rewrite<= *-inverses_binary64 (/.f64 t t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 y z) t) t))): 51 points increase in error, 0 points decrease in error

   if -1e-242 < (/.f64 y z) < 1.0000000000000001e-253

   1. Initial program 18.9

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

   2. Simplified 14.1

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Proof
      (*.f64 x (/.f64 y z)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= /-rgt-identity_binary64 (/.f64 (/.f64 y z) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (/.f64 (/.f64 y z) (Rewrite<= *-inverses_binary64 (/.f64 t t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 y z) t) t))): 51 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 0.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+256}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-242}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-253}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1360

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+256}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -5 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-253}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	0.5
Cost	1360

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := \frac{y \cdot x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+256}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-253}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	6.3
Cost	320

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

Reproduce

herbie shell --seed 2022325 
(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)))