Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Average Error: 11.3 → 2.3

Time: 9.4s

Precision: binary64

Cost: 840

Math

\[\frac{x \cdot \left(y - z\right)}{t - z} \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-151}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.4e-151)
   (* x (/ (- z y) (- z t)))
   (if (<= z 7.6e-252) (* (/ x (- t z)) (- y z)) (/ x (/ (- t z) (- y z))))))

C

double code(double x, double y, double z, double t) {
	return (x * (y - z)) / (t - z);
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.4e-151) {
		tmp = x * ((z - y) / (z - t));
	} else if (z <= 7.6e-252) {
		tmp = (x / (t - z)) * (y - z);
	} else {
		tmp = x / ((t - z) / (y - 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 - z)
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 (z <= (-1.4d-151)) then
        tmp = x * ((z - y) / (z - t))
    else if (z <= 7.6d-252) then
        tmp = (x / (t - z)) * (y - z)
    else
        tmp = x / ((t - z) / (y - 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 - z);
}

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.4e-151) {
		tmp = x * ((z - y) / (z - t));
	} else if (z <= 7.6e-252) {
		tmp = (x / (t - z)) * (y - z);
	} else {
		tmp = x / ((t - z) / (y - z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	tmp = 0
	if z <= -1.4e-151:
		tmp = x * ((z - y) / (z - t))
	elif z <= 7.6e-252:
		tmp = (x / (t - z)) * (y - z)
	else:
		tmp = x / ((t - z) / (y - z))
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(y - z)) / Float64(t - z))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.4e-151)
		tmp = Float64(x * Float64(Float64(z - y) / Float64(z - t)));
	elseif (z <= 7.6e-252)
		tmp = Float64(Float64(x / Float64(t - z)) * Float64(y - z));
	else
		tmp = Float64(x / Float64(Float64(t - z) / Float64(y - z)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * (y - z)) / (t - z);
end

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.4e-151)
		tmp = x * ((z - y) / (z - t));
	elseif (z <= 7.6e-252)
		tmp = (x / (t - z)) * (y - z);
	else
		tmp = x / ((t - z) / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[z, -1.4e-151], N[(x * N[(N[(z - y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.6e-252], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(t - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	11.3
Target	2.2
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

2. if z < -1.4e-151

   1. Initial program 13.0

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]

   2. Simplified 1.0

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

      [Start] 13.0	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-*r/ [<=] 1.0	\[ \color{blue}{x \cdot \frac{y - z}{t - z}} \]
      sub-neg [=>] 1.0	\[ x \cdot \frac{\color{blue}{y + \left(-z\right)}}{t - z} \]
      +-commutative [=>] 1.0	\[ x \cdot \frac{\color{blue}{\left(-z\right) + y}}{t - z} \]
      neg-sub0 [=>] 1.0	\[ x \cdot \frac{\color{blue}{\left(0 - z\right)} + y}{t - z} \]
      associate-+l- [=>] 1.0	\[ x \cdot \frac{\color{blue}{0 - \left(z - y\right)}}{t - z} \]
      sub0-neg [=>] 1.0	\[ x \cdot \frac{\color{blue}{-\left(z - y\right)}}{t - z} \]
      neg-mul-1 [=>] 1.0	\[ x \cdot \frac{\color{blue}{-1 \cdot \left(z - y\right)}}{t - z} \]
      sub-neg [=>] 1.0	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{t + \left(-z\right)}} \]
      +-commutative [=>] 1.0	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(-z\right) + t}} \]
      neg-sub0 [=>] 1.0	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(0 - z\right)} + t} \]
      associate-+l- [=>] 1.0	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{0 - \left(z - t\right)}} \]
      sub0-neg [=>] 1.0	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-\left(z - t\right)}} \]
      neg-mul-1 [=>] 1.0	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-1 \cdot \left(z - t\right)}} \]
      times-frac [=>] 1.0	\[ x \cdot \color{blue}{\left(\frac{-1}{-1} \cdot \frac{z - y}{z - t}\right)} \]
      metadata-eval [=>] 1.0	\[ x \cdot \left(\color{blue}{1} \cdot \frac{z - y}{z - t}\right) \]
      *-lft-identity [=>] 1.0	\[ x \cdot \color{blue}{\frac{z - y}{z - t}} \]

   if -1.4e-151 < z < 7.6e-252

   1. Initial program 6.1

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]

   2. Simplified 6.6

      \[\leadsto \color{blue}{\frac{x}{t - z} \cdot \left(y - z\right)} \]
      Proof

      [Start] 6.1	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-*l/ [<=] 6.6	\[ \color{blue}{\frac{x}{t - z} \cdot \left(y - z\right)} \]

   if 7.6e-252 < z

   1. Initial program 11.7

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]

   2. Simplified 1.9

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

      [Start] 11.7	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-/l* [=>] 1.9	\[ \color{blue}{\frac{x}{\frac{t - z}{y - z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-151}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.8
Cost	1372

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ t_2 := y \cdot \frac{x}{t - z}\\ t_3 := \frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+210}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.45 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-50}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	19.6
Cost	1108

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-252}:\\ \;\;\;\;\frac{y}{\frac{t}{x}}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	2.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-151} \lor \neg \left(z \leq 4.5 \cdot 10^{-217}\right):\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \end{array} \]

Alternative 4
Error	26.4
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-96}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	17.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-20} \lor \neg \left(z \leq 2.1 \cdot 10^{-113}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 6
Error	17.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-113}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \end{array} \]

Alternative 7
Error	25.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	24.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.0032:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	2.2
Cost	576

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

Alternative 10
Error	40.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (/ x (/ (- t z) (- y z)))

  (/ (* x (- y z)) (- t z)))