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

Percentage Accurate: 81.9% → 95.7%

Time: 4.2s

Precision: binary64

Cost: 1353

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

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{y}{z} \cdot t}{t}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-216} \lor \neg \left(t_1 \leq 4 \cdot 10^{-323}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

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) t) t)))
   (if (or (<= t_1 -5e-216) (not (<= t_1 4e-323)))
     (/ x (/ z y))
     (* 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) * t) / t;
	double tmp;
	if ((t_1 <= -5e-216) || !(t_1 <= 4e-323)) {
		tmp = x / (z / y);
	} 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) * t) / t
    if ((t_1 <= (-5d-216)) .or. (.not. (t_1 <= 4d-323))) then
        tmp = x / (z / y)
    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) * t) / t;
	double tmp;
	if ((t_1 <= -5e-216) || !(t_1 <= 4e-323)) {
		tmp = x / (z / y);
	} 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) * t) / t
	tmp = 0
	if (t_1 <= -5e-216) or not (t_1 <= 4e-323):
		tmp = x / (z / y)
	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(Float64(y / z) * t) / t)
	tmp = 0.0
	if ((t_1 <= -5e-216) || !(t_1 <= 4e-323))
		tmp = Float64(x / Float64(z / y));
	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) * t) / t;
	tmp = 0.0;
	if ((t_1 <= -5e-216) || ~((t_1 <= 4e-323)))
		tmp = x / (z / y);
	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[(N[(y / z), $MachinePrecision] * t), $MachinePrecision] / t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e-216], N[Not[LessEqual[t$95$1, 4e-323]], $MachinePrecision]], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	81.9%
Target	98.0%
Herbie	95.7%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (/.f64 y z) t) t) < -5.00000000000000021e-216 or 3.95253e-323 < (/.f64 (*.f64 (/.f64 y z) t) t)

   1. Initial program 93.5%

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

   2. Simplified 97.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Step-by-step derivation

      [Start] 93.5%	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 97.9%	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 97.9%	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 97.9%	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
      Step-by-step derivation

      [Start] 97.9%	\[ x \cdot \frac{y}{z} \]
      clear-num [=>] 97.8%	\[ x \cdot \color{blue}{\frac{1}{\frac{z}{y}}} \]
      un-div-inv [=>] 98.3%	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

   if -5.00000000000000021e-216 < (/.f64 (*.f64 (/.f64 y z) t) t) < 3.95253e-323

   1. Initial program 52.2%

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

   2. Simplified 97.6%

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]
      Step-by-step derivation

      [Start] 52.2%	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      *-commutative [=>] 52.2%	\[ \color{blue}{\frac{\frac{y}{z} \cdot t}{t} \cdot x} \]
      associate-/l* [=>] 73.6%	\[ \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \cdot x \]
      *-inverses [=>] 73.6%	\[ \frac{\frac{y}{z}}{\color{blue}{1}} \cdot x \]
      /-rgt-identity [=>] 73.6%	\[ \color{blue}{\frac{y}{z}} \cdot x \]
      associate-*l/ [=>] 98.2%	\[ \color{blue}{\frac{y \cdot x}{z}} \]
      associate-*r/ [<=] 97.6%	\[ \color{blue}{y \cdot \frac{x}{z}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \leq -5 \cdot 10^{-216} \lor \neg \left(\frac{\frac{y}{z} \cdot t}{t} \leq 4 \cdot 10^{-323}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.7%
Cost	1353

\[\begin{array}{l} t_1 := \frac{\frac{y}{z} \cdot t}{t}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-216} \lor \neg \left(t_1 \leq 4 \cdot 10^{-323}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	92.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-143}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 3
Accuracy	92.1%
Cost	320

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

Reproduce

herbie shell --seed 2023167 
(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)))