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

Average Error: 11.8 → 0.4

Time: 15.5s

Precision: binary64

Cost: 3153

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-280} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq 5 \cdot 10^{+257}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))))
   (if (<= t_1 -5e+303)
     (/ x (/ (- t z) (- y z)))
     (if (or (<= t_1 -1e-280) (and (not (<= t_1 0.0)) (<= t_1 5e+257)))
       t_1
       (* x (/ (- z y) (- z t)))))))

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 t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -5e+303) {
		tmp = x / ((t - z) / (y - z));
	} else if ((t_1 <= -1e-280) || (!(t_1 <= 0.0) && (t_1 <= 5e+257))) {
		tmp = t_1;
	} else {
		tmp = x * ((z - y) / (z - t));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (x * (y - z)) / (t - z)
    if (t_1 <= (-5d+303)) then
        tmp = x / ((t - z) / (y - z))
    else if ((t_1 <= (-1d-280)) .or. (.not. (t_1 <= 0.0d0)) .and. (t_1 <= 5d+257)) then
        tmp = t_1
    else
        tmp = x * ((z - y) / (z - t))
    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 t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= -5e+303) {
		tmp = x / ((t - z) / (y - z));
	} else if ((t_1 <= -1e-280) || (!(t_1 <= 0.0) && (t_1 <= 5e+257))) {
		tmp = t_1;
	} else {
		tmp = x * ((z - y) / (z - t));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = (x * (y - z)) / (t - z)
	tmp = 0
	if t_1 <= -5e+303:
		tmp = x / ((t - z) / (y - z))
	elif (t_1 <= -1e-280) or (not (t_1 <= 0.0) and (t_1 <= 5e+257)):
		tmp = t_1
	else:
		tmp = x * ((z - y) / (z - t))
	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)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= -5e+303)
		tmp = Float64(x / Float64(Float64(t - z) / Float64(y - z)));
	elseif ((t_1 <= -1e-280) || (!(t_1 <= 0.0) && (t_1 <= 5e+257)))
		tmp = t_1;
	else
		tmp = Float64(x * Float64(Float64(z - y) / Float64(z - t)));
	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)
	t_1 = (x * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_1 <= -5e+303)
		tmp = x / ((t - z) / (y - z));
	elseif ((t_1 <= -1e-280) || (~((t_1 <= 0.0)) && (t_1 <= 5e+257)))
		tmp = t_1;
	else
		tmp = x * ((z - y) / (z - t));
	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_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+303], N[(x / N[(N[(t - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, -1e-280], And[N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision], LessEqual[t$95$1, 5e+257]]], t$95$1, N[(x * N[(N[(z - y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	11.8
Target	2.1
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -4.9999999999999997e303

   1. Initial program 63.2

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

   2. Simplified 0.2

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

      [Start] 63.2	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-/l* [=>] 0.2	\[ \color{blue}{\frac{x}{\frac{t - z}{y - z}}} \]

   if -4.9999999999999997e303 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -9.9999999999999996e-281 or -0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 5.00000000000000028e257

   1. Initial program 0.3

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

   if -9.9999999999999996e-281 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -0.0 or 5.00000000000000028e257 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 29.7

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

   2. Simplified 0.9

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

      [Start] 29.7	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-*r/ [<=] 0.9	\[ \color{blue}{x \cdot \frac{y - z}{t - z}} \]
      sub-neg [=>] 0.9	\[ x \cdot \frac{\color{blue}{y + \left(-z\right)}}{t - z} \]
      +-commutative [=>] 0.9	\[ x \cdot \frac{\color{blue}{\left(-z\right) + y}}{t - z} \]
      neg-sub0 [=>] 0.9	\[ x \cdot \frac{\color{blue}{\left(0 - z\right)} + y}{t - z} \]
      associate-+l- [=>] 0.9	\[ x \cdot \frac{\color{blue}{0 - \left(z - y\right)}}{t - z} \]
      sub0-neg [=>] 0.9	\[ x \cdot \frac{\color{blue}{-\left(z - y\right)}}{t - z} \]
      neg-mul-1 [=>] 0.9	\[ x \cdot \frac{\color{blue}{-1 \cdot \left(z - y\right)}}{t - z} \]
      sub-neg [=>] 0.9	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{t + \left(-z\right)}} \]
      +-commutative [=>] 0.9	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(-z\right) + t}} \]
      neg-sub0 [=>] 0.9	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(0 - z\right)} + t} \]
      associate-+l- [=>] 0.9	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{0 - \left(z - t\right)}} \]
      sub0-neg [=>] 0.9	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-\left(z - t\right)}} \]
      neg-mul-1 [=>] 0.9	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-1 \cdot \left(z - t\right)}} \]
      times-frac [=>] 0.9	\[ x \cdot \color{blue}{\left(\frac{-1}{-1} \cdot \frac{z - y}{z - t}\right)} \]
      metadata-eval [=>] 0.9	\[ x \cdot \left(\color{blue}{1} \cdot \frac{z - y}{z - t}\right) \]
      *-lft-identity [=>] 0.9	\[ x \cdot \color{blue}{\frac{z - y}{z - t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -1 \cdot 10^{-280} \lor \neg \left(\frac{x \cdot \left(y - z\right)}{t - z} \leq 0\right) \land \frac{x \cdot \left(y - z\right)}{t - z} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.3
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ t_2 := y \cdot \frac{x}{t - z}\\ t_3 := \frac{x}{\frac{t - z}{y}}\\ \mathbf{if}\;y \leq -9 \cdot 10^{-5}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+83}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+137}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	17.4
Cost	1240

\[\begin{array}{l} t_1 := \frac{y - z}{\frac{t}{x}}\\ t_2 := x \cdot \left(1 - \frac{y}{z}\right)\\ t_3 := \frac{x}{\frac{t - z}{y}}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-214}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+40}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{1 - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	27.0
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+121}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	27.1
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+121}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	27.1
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+121}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{-z}{\frac{t}{x}}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	27.1
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+121}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+35}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	2.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-46} \lor \neg \left(z \leq 4.5 \cdot 10^{-128}\right):\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 8
Error	2.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-46} \lor \neg \left(z \leq 6.6 \cdot 10^{-109}\right):\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 9
Error	19.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-64} \lor \neg \left(z \leq 6.8 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \end{array} \]

Alternative 10
Error	16.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.46 \cdot 10^{-62} \lor \neg \left(z \leq 2.35 \cdot 10^{+40}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 11
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \end{array} \]

Alternative 12
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{1 - \frac{t}{z}}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+35}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \end{array} \]

Alternative 13
Error	2.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+213}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+271}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \end{array} \]

Alternative 14
Error	38.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{-175}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	26.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -3.35 \cdot 10^{-47}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	40.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023039 
(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)))