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

Average Error: 22.62% → 1.71%

Time: 4.8s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+287}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-182}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

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) -5e+287)
     (/ (* y x) z)
     (if (<= (/ y z) -1e-133)
       t_1
       (if (<= (/ y z) 1e-182)
         (/ y (/ z x))
         (if (<= (/ y z) 2e+82) t_1 (* y (/ x z))))))))

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) <= -5e+287) {
		tmp = (y * x) / z;
	} else if ((y / z) <= -1e-133) {
		tmp = t_1;
	} else if ((y / z) <= 1e-182) {
		tmp = y / (z / x);
	} else if ((y / z) <= 2e+82) {
		tmp = t_1;
	} else {
		tmp = y * (x / z);
	}
	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) <= (-5d+287)) then
        tmp = (y * x) / z
    else if ((y / z) <= (-1d-133)) then
        tmp = t_1
    else if ((y / z) <= 1d-182) then
        tmp = y / (z / x)
    else if ((y / z) <= 2d+82) then
        tmp = t_1
    else
        tmp = y * (x / z)
    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) <= -5e+287) {
		tmp = (y * x) / z;
	} else if ((y / z) <= -1e-133) {
		tmp = t_1;
	} else if ((y / z) <= 1e-182) {
		tmp = y / (z / x);
	} else if ((y / z) <= 2e+82) {
		tmp = t_1;
	} else {
		tmp = y * (x / z);
	}
	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) <= -5e+287:
		tmp = (y * x) / z
	elif (y / z) <= -1e-133:
		tmp = t_1
	elif (y / z) <= 1e-182:
		tmp = y / (z / x)
	elif (y / z) <= 2e+82:
		tmp = t_1
	else:
		tmp = y * (x / z)
	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)
	tmp = 0.0
	if (Float64(y / z) <= -5e+287)
		tmp = Float64(Float64(y * x) / z);
	elseif (Float64(y / z) <= -1e-133)
		tmp = t_1;
	elseif (Float64(y / z) <= 1e-182)
		tmp = Float64(y / Float64(z / x));
	elseif (Float64(y / z) <= 2e+82)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(x / z));
	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) <= -5e+287)
		tmp = (y * x) / z;
	elseif ((y / z) <= -1e-133)
		tmp = t_1;
	elseif ((y / z) <= 1e-182)
		tmp = y / (z / x);
	elseif ((y / z) <= 2e+82)
		tmp = t_1;
	else
		tmp = y * (x / z);
	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]}, If[LessEqual[N[(y / z), $MachinePrecision], -5e+287], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], -1e-133], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], 1e-182], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y / z), $MachinePrecision], 2e+82], t$95$1, N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	22.62%
Target	2.34%
Herbie	1.71%

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 y z) < -5e287

   1. Initial program 87.34

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

   2. Simplified 79.01

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

      [Start] 87.34	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 79.01	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 79.01	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 79.01	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 0.34

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

   if -5e287 < (/.f64 y z) < -1.0000000000000001e-133 or 1e-182 < (/.f64 y z) < 1.9999999999999999e82

   1. Initial program 12.04

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

   2. Simplified 0.39

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

      [Start] 12.04	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 0.39	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 0.39	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 0.39	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   if -1.0000000000000001e-133 < (/.f64 y z) < 1e-182

   1. Initial program 25.28

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

   2. Simplified 1.94

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

      [Start] 25.28	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 12.16	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 12.16	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 12.16	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 2.14	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 1.94	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 1.94	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   3. Applied egg-rr 1.93

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

   if 1.9999999999999999e82 < (/.f64 y z)

   1. Initial program 41.73

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

   2. Simplified 6.46

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

      [Start] 41.73	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 20.86	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 20.86	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 20.86	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 7.21	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 6.46	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 6.46	\[ \color{blue}{y \cdot \frac{x}{z}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 1.71

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+287}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-133}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-182}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.52%
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+213} \lor \neg \left(\frac{y}{z} \leq -5 \cdot 10^{-171}\right) \land \left(\frac{y}{z} \leq 0 \lor \neg \left(\frac{y}{z} \leq 2 \cdot 10^{+82}\right)\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Error	1.77%
Cost	1360

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -5 \cdot 10^{+213}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 10^{-182}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	9.34%
Cost	320

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

Reproduce

herbie shell --seed 2023088 
(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)))