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

Average Accuracy: 81.3% → 98.5%

Time: 3.8s

Precision: binary64

Cost: 1360

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{y}}\\ t_2 := \frac{y \cdot x}{z}\\ \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+203}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{y}{z} \leq 4 \cdot 10^{-313}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (/ z y))) (t_2 (/ (* y x) z)))
   (if (<= (/ y z) -1e+203)
     t_2
     (if (<= (/ y z) -1e-102)
       t_1
       (if (<= (/ y z) 4e-313)
         t_2
         (if (<= (/ y z) 2e+126) t_1 (/ y (/ z 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 t_1 = x / (z / y);
	double t_2 = (y * x) / z;
	double tmp;
	if ((y / z) <= -1e+203) {
		tmp = t_2;
	} else if ((y / z) <= -1e-102) {
		tmp = t_1;
	} else if ((y / z) <= 4e-313) {
		tmp = t_2;
	} else if ((y / z) <= 2e+126) {
		tmp = t_1;
	} else {
		tmp = y / (z / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / (z / y)
    t_2 = (y * x) / z
    if ((y / z) <= (-1d+203)) then
        tmp = t_2
    else if ((y / z) <= (-1d-102)) then
        tmp = t_1
    else if ((y / z) <= 4d-313) then
        tmp = t_2
    else if ((y / z) <= 2d+126) then
        tmp = t_1
    else
        tmp = y / (z / 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 t_1 = x / (z / y);
	double t_2 = (y * x) / z;
	double tmp;
	if ((y / z) <= -1e+203) {
		tmp = t_2;
	} else if ((y / z) <= -1e-102) {
		tmp = t_1;
	} else if ((y / z) <= 4e-313) {
		tmp = t_2;
	} else if ((y / z) <= 2e+126) {
		tmp = t_1;
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = x / (z / y)
	t_2 = (y * x) / z
	tmp = 0
	if (y / z) <= -1e+203:
		tmp = t_2
	elif (y / z) <= -1e-102:
		tmp = t_1
	elif (y / z) <= 4e-313:
		tmp = t_2
	elif (y / z) <= 2e+126:
		tmp = t_1
	else:
		tmp = y / (z / 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)
	t_1 = Float64(x / Float64(z / y))
	t_2 = Float64(Float64(y * x) / z)
	tmp = 0.0
	if (Float64(y / z) <= -1e+203)
		tmp = t_2;
	elseif (Float64(y / z) <= -1e-102)
		tmp = t_1;
	elseif (Float64(y / z) <= 4e-313)
		tmp = t_2;
	elseif (Float64(y / z) <= 2e+126)
		tmp = t_1;
	else
		tmp = Float64(y / Float64(z / 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)
	t_1 = x / (z / y);
	t_2 = (y * x) / z;
	tmp = 0.0;
	if ((y / z) <= -1e+203)
		tmp = t_2;
	elseif ((y / z) <= -1e-102)
		tmp = t_1;
	elseif ((y / z) <= 4e-313)
		tmp = t_2;
	elseif ((y / z) <= 2e+126)
		tmp = t_1;
	else
		tmp = y / (z / 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_] := Block[{t$95$1 = N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(y / z), $MachinePrecision], -1e+203], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], -1e-102], t$95$1, If[LessEqual[N[(y / z), $MachinePrecision], 4e-313], t$95$2, If[LessEqual[N[(y / z), $MachinePrecision], 2e+126], t$95$1, N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	81.3%
Target	98.2%
Herbie	98.5%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 y z) < -9.9999999999999999e202 or -9.99999999999999933e-103 < (/.f64 y z) < 4.0000000000037e-313

   1. Initial program 81.4%

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

   2. Simplified 93.7%

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

      [Start] 81.4	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 93.7	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 93.7	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 93.7	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 97.7%

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

   if -9.9999999999999999e202 < (/.f64 y z) < -9.99999999999999933e-103 or 4.0000000000037e-313 < (/.f64 y z) < 1.99999999999999985e126

   1. Initial program 88.3%

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

   2. Simplified 99.5%

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

      [Start] 88.3	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 99.5	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 99.5	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 99.5	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Taylor expanded in x around 0 85.5%

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

   4. Simplified 99.0%

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

      [Start] 85.5	\[ \frac{y \cdot x}{z} \]
      *-commutative [=>] 85.5	\[ \frac{\color{blue}{x \cdot y}}{z} \]
      associate-/l* [=>] 99.0	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

   if 1.99999999999999985e126 < (/.f64 y z)

   1. Initial program 57.9%

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

   2. Simplified 97.4%

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

      [Start] 57.9	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      *-commutative [=>] 57.9	\[ \color{blue}{\frac{\frac{y}{z} \cdot t}{t} \cdot x} \]
      associate-/l* [=>] 77.7	\[ \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \cdot x \]
      *-inverses [=>] 77.7	\[ \frac{\frac{y}{z}}{\color{blue}{1}} \cdot x \]
      /-rgt-identity [=>] 77.7	\[ \color{blue}{\frac{y}{z}} \cdot x \]
      associate-*l/ [=>] 98.0	\[ \color{blue}{\frac{y \cdot x}{z}} \]
      associate-*r/ [<=] 97.4	\[ \color{blue}{y \cdot \frac{x}{z}} \]

   3. Applied egg-rr 97.5%

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

      [Start] 97.4	\[ y \cdot \frac{x}{z} \]
      clear-num [=>] 97.4	\[ y \cdot \color{blue}{\frac{1}{\frac{z}{x}}} \]
      un-div-inv [=>] 97.5	\[ \color{blue}{\frac{y}{\frac{z}{x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+203}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \leq 4 \cdot 10^{-313}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{+286} \lor \neg \left(\frac{y}{z} \leq -1 \cdot 10^{-197}\right) \land \left(\frac{y}{z} \leq 4 \cdot 10^{-272} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{+165}\right)\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty \lor \neg \left(\frac{y}{z} \leq -1 \cdot 10^{-181} \lor \neg \left(\frac{y}{z} \leq 4 \cdot 10^{-313}\right) \land \frac{y}{z} \leq 5 \cdot 10^{+165}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	1361

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -2 \cdot 10^{-213} \lor \neg \left(\frac{y}{z} \leq 4 \cdot 10^{-313}\right) \land \frac{y}{z} \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 4
Accuracy	92.3%
Cost	320

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

Reproduce

herbie shell --seed 2023160 
(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)))