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

Average Error: 14.7 → 0.9

Time: 4.4s

Precision: binary64

Cost: 1488

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+113}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-216}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(y \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ y z) -1e+113)
   (* y (/ x z))
   (if (<= (/ y z) -1e-303)
     (/ x (/ z y))
     (if (<= (/ y z) 1e-216)
       (/ y (/ z x))
       (if (<= (/ y z) 2e+134) (* (/ y z) x) (* (/ 1.0 z) (* y x)))))))

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 tmp;
	if ((y / z) <= -1e+113) {
		tmp = y * (x / z);
	} else if ((y / z) <= -1e-303) {
		tmp = x / (z / y);
	} else if ((y / z) <= 1e-216) {
		tmp = y / (z / x);
	} else if ((y / z) <= 2e+134) {
		tmp = (y / z) * x;
	} else {
		tmp = (1.0 / z) * (y * x);
	}
	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) :: tmp
    if ((y / z) <= (-1d+113)) then
        tmp = y * (x / z)
    else if ((y / z) <= (-1d-303)) then
        tmp = x / (z / y)
    else if ((y / z) <= 1d-216) then
        tmp = y / (z / x)
    else if ((y / z) <= 2d+134) then
        tmp = (y / z) * x
    else
        tmp = (1.0d0 / z) * (y * x)
    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 tmp;
	if ((y / z) <= -1e+113) {
		tmp = y * (x / z);
	} else if ((y / z) <= -1e-303) {
		tmp = x / (z / y);
	} else if ((y / z) <= 1e-216) {
		tmp = y / (z / x);
	} else if ((y / z) <= 2e+134) {
		tmp = (y / z) * x;
	} else {
		tmp = (1.0 / z) * (y * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x * (((y / z) * t) / t)

↓

def code(x, y, z, t):
	tmp = 0
	if (y / z) <= -1e+113:
		tmp = y * (x / z)
	elif (y / z) <= -1e-303:
		tmp = x / (z / y)
	elif (y / z) <= 1e-216:
		tmp = y / (z / x)
	elif (y / z) <= 2e+134:
		tmp = (y / z) * x
	else:
		tmp = (1.0 / z) * (y * x)
	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)
	tmp = 0.0
	if (Float64(y / z) <= -1e+113)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(y / z) <= -1e-303)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(y / z) <= 1e-216)
		tmp = Float64(y / Float64(z / x));
	elseif (Float64(y / z) <= 2e+134)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(Float64(1.0 / z) * Float64(y * x));
	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)
	tmp = 0.0;
	if ((y / z) <= -1e+113)
		tmp = y * (x / z);
	elseif ((y / z) <= -1e-303)
		tmp = x / (z / y);
	elseif ((y / z) <= 1e-216)
		tmp = y / (z / x);
	elseif ((y / z) <= 2e+134)
		tmp = (y / z) * x;
	else
		tmp = (1.0 / z) * (y * x);
	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_] := If[LessEqual[N[(y / z), $MachinePrecision], -1e+113], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], -1e-303], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 1e-216], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 2e+134], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	14.7
Target	1.6
Herbie	0.9

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

Derivation

1. Split input into 5 regimes

2. if (/.f64 y z) < -1e113

   1. Initial program 31.3

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

   2. Simplified 4.3

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]
      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)): 2 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) z): 0 points increase in error, 3 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z))): 4 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, 4 points decrease in error
      (*.f64 x (/.f64 (/.f64 y z) (Rewrite<= *-inverses_binary64 (/.f64 t t)))): 3 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 y z) t) t))): 0 points increase in error, 3 points decrease in error

   if -1e113 < (/.f64 y z) < -9.99999999999999931e-304

   1. Initial program 7.5

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      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)): 2 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) z): 0 points increase in error, 3 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z))): 4 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, 4 points decrease in error
      (*.f64 x (/.f64 (/.f64 y z) (Rewrite<= *-inverses_binary64 (/.f64 t t)))): 3 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 y z) t) t))): 0 points increase in error, 3 points decrease in error

   3. Applied egg-rr 0.2

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

   if -9.99999999999999931e-304 < (/.f64 y z) < 1e-216

   1. Initial program 18.2

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

   2. Simplified 0.4

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]
      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)): 2 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) z): 0 points increase in error, 3 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z))): 4 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, 4 points decrease in error
      (*.f64 x (/.f64 (/.f64 y z) (Rewrite<= *-inverses_binary64 (/.f64 t t)))): 3 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 y z) t) t))): 0 points increase in error, 3 points decrease in error

   3. Applied egg-rr 0.4

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

   if 1e-216 < (/.f64 y z) < 1.99999999999999984e134

   1. Initial program 7.4

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      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)): 2 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) z): 0 points increase in error, 3 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z))): 4 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, 4 points decrease in error
      (*.f64 x (/.f64 (/.f64 y z) (Rewrite<= *-inverses_binary64 (/.f64 t t)))): 3 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 y z) t) t))): 0 points increase in error, 3 points decrease in error

   if 1.99999999999999984e134 < (/.f64 y z)

   1. Initial program 32.4

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

   2. Simplified 18.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      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)): 2 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) z): 0 points increase in error, 3 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 y z))): 4 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, 4 points decrease in error
      (*.f64 x (/.f64 (/.f64 y z) (Rewrite<= *-inverses_binary64 (/.f64 t t)))): 3 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 y z) t) t))): 0 points increase in error, 3 points decrease in error

   3. Taylor expanded in x around 0 2.3

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

   4. Applied egg-rr 2.4

      \[\leadsto \color{blue}{\frac{1}{z} \cdot \left(y \cdot x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+113}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-216}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+113} \lor \neg \left(\frac{y}{z} \leq -4 \cdot 10^{-159} \lor \neg \left(\frac{y}{z} \leq 10^{-275}\right) \land \frac{y}{z} \leq 2 \cdot 10^{+222}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Error	1.0
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+113} \lor \neg \left(\frac{y}{z} \leq -4 \cdot 10^{-159} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{-321}\right) \land \frac{y}{z} \leq 5 \cdot 10^{+209}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	1360

\[\begin{array}{l} t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-216}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	6.5
Cost	320

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

Reproduce

herbie shell --seed 2022343 
(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)))