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

Average Accuracy: 77.7% → 97.3%

Time: 4.4s

Precision: binary64

Cost: 1101

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{-256} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{-160}\right) \land \frac{y}{z} \leq 4 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ y z) -2e-256)
         (and (not (<= (/ y z) 5e-160)) (<= (/ y z) 4e+178)))
   (/ 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 tmp;
	if (((y / z) <= -2e-256) || (!((y / z) <= 5e-160) && ((y / z) <= 4e+178))) {
		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) :: tmp
    if (((y / z) <= (-2d-256)) .or. (.not. ((y / z) <= 5d-160)) .and. ((y / z) <= 4d+178)) 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 tmp;
	if (((y / z) <= -2e-256) || (!((y / z) <= 5e-160) && ((y / z) <= 4e+178))) {
		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):
	tmp = 0
	if ((y / z) <= -2e-256) or (not ((y / z) <= 5e-160) and ((y / z) <= 4e+178)):
		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)
	tmp = 0.0
	if ((Float64(y / z) <= -2e-256) || (!(Float64(y / z) <= 5e-160) && (Float64(y / z) <= 4e+178)))
		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)
	tmp = 0.0;
	if (((y / z) <= -2e-256) || (~(((y / z) <= 5e-160)) && ((y / z) <= 4e+178)))
		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_] := If[Or[LessEqual[N[(y / z), $MachinePrecision], -2e-256], And[N[Not[LessEqual[N[(y / z), $MachinePrecision], 5e-160]], $MachinePrecision], LessEqual[N[(y / z), $MachinePrecision], 4e+178]]], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Target

Original	77.7%
Target	97.7%
Herbie	97.3%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 y z) < -1.99999999999999995e-256 or 4.99999999999999994e-160 < (/.f64 y z) < 4.0000000000000002e178

   1. Initial program 83.3%

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

   2. Simplified 96.4%

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

      [Start] 83.3	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 96.4	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 96.4	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 96.4	\[ x \cdot \color{blue}{\frac{y}{z}} \]

   3. Applied egg-rr 96.7%

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

      [Start] 96.4	\[ x \cdot \frac{y}{z} \]
      associate-*r/ [=>] 85.3	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-/l* [=>] 96.7	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

   if -1.99999999999999995e-256 < (/.f64 y z) < 4.99999999999999994e-160 or 4.0000000000000002e178 < (/.f64 y z)

   1. Initial program 67.1%

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

   2. Simplified 98.3%

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

      [Start] 67.1	\[ x \cdot \frac{\frac{y}{z} \cdot t}{t} \]
      associate-/l* [=>] 79.5	\[ x \cdot \color{blue}{\frac{\frac{y}{z}}{\frac{t}{t}}} \]
      *-inverses [=>] 79.5	\[ x \cdot \frac{\frac{y}{z}}{\color{blue}{1}} \]
      /-rgt-identity [=>] 79.5	\[ x \cdot \color{blue}{\frac{y}{z}} \]
      associate-*r/ [=>] 98.8	\[ \color{blue}{\frac{x \cdot y}{z}} \]
      associate-*l/ [<=] 98.3	\[ \color{blue}{\frac{x}{z} \cdot y} \]
      *-commutative [<=] 98.3	\[ \color{blue}{y \cdot \frac{x}{z}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2 \cdot 10^{-256} \lor \neg \left(\frac{y}{z} \leq 5 \cdot 10^{-160}\right) \land \frac{y}{z} \leq 4 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.7%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -1 \cdot 10^{-153} \lor \neg \left(\frac{y}{z} \leq 4 \cdot 10^{-165}\right) \land \frac{y}{z} \leq 4 \cdot 10^{+178}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	90.6%
Cost	320

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

Reproduce

herbie shell --seed 2023133 
(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)))