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

Average Error: 18.46% → 2.45%

Time: 14.7s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-283}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- y z)) (- t z))))
   (if (<= t_1 5e-283)
     (* x (/ (- z y) (- z t)))
     (if (<= t_1 2e+285) t_1 (/ 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 t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= 5e-283) {
		tmp = x * ((z - y) / (z - t));
	} else if (t_1 <= 2e+285) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (x * (y - z)) / (t - z)
    if (t_1 <= 5d-283) then
        tmp = x * ((z - y) / (z - t))
    else if (t_1 <= 2d+285) then
        tmp = t_1
    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 t_1 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_1 <= 5e-283) {
		tmp = x * ((z - y) / (z - t));
	} else if (t_1 <= 2e+285) {
		tmp = t_1;
	} 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):
	t_1 = (x * (y - z)) / (t - z)
	tmp = 0
	if t_1 <= 5e-283:
		tmp = x * ((z - y) / (z - t))
	elif t_1 <= 2e+285:
		tmp = t_1
	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)
	t_1 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_1 <= 5e-283)
		tmp = Float64(x * Float64(Float64(z - y) / Float64(z - t)));
	elseif (t_1 <= 2e+285)
		tmp = t_1;
	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)
	t_1 = (x * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_1 <= 5e-283)
		tmp = x * ((z - y) / (z - t));
	elseif (t_1 <= 2e+285)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-283], N[(x * N[(N[(z - y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+285], t$95$1, N[(x / N[(N[(t - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	18.46%
Target	3.6%
Herbie	2.45%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 5.0000000000000001e-283

   1. Initial program 17.63

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

   2. Simplified 3.72

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

      [Start] 17.63	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-*r/ [<=] 3.72	\[ \color{blue}{x \cdot \frac{y - z}{t - z}} \]
      sub-neg [=>] 3.72	\[ x \cdot \frac{\color{blue}{y + \left(-z\right)}}{t - z} \]
      +-commutative [=>] 3.72	\[ x \cdot \frac{\color{blue}{\left(-z\right) + y}}{t - z} \]
      neg-sub0 [=>] 3.72	\[ x \cdot \frac{\color{blue}{\left(0 - z\right)} + y}{t - z} \]
      associate-+l- [=>] 3.72	\[ x \cdot \frac{\color{blue}{0 - \left(z - y\right)}}{t - z} \]
      sub0-neg [=>] 3.72	\[ x \cdot \frac{\color{blue}{-\left(z - y\right)}}{t - z} \]
      neg-mul-1 [=>] 3.72	\[ x \cdot \frac{\color{blue}{-1 \cdot \left(z - y\right)}}{t - z} \]
      sub-neg [=>] 3.72	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{t + \left(-z\right)}} \]
      +-commutative [=>] 3.72	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(-z\right) + t}} \]
      neg-sub0 [=>] 3.72	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(0 - z\right)} + t} \]
      associate-+l- [=>] 3.72	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{0 - \left(z - t\right)}} \]
      sub0-neg [=>] 3.72	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-\left(z - t\right)}} \]
      neg-mul-1 [=>] 3.72	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-1 \cdot \left(z - t\right)}} \]
      times-frac [=>] 3.72	\[ x \cdot \color{blue}{\left(\frac{-1}{-1} \cdot \frac{z - y}{z - t}\right)} \]
      metadata-eval [=>] 3.72	\[ x \cdot \left(\color{blue}{1} \cdot \frac{z - y}{z - t}\right) \]
      *-lft-identity [=>] 3.72	\[ x \cdot \color{blue}{\frac{z - y}{z - t}} \]

   if 5.0000000000000001e-283 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 2e285

   1. Initial program 0.6

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

   if 2e285 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 93.53

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

   2. Simplified 1.62

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

      [Start] 93.53	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-/l* [=>] 1.62	\[ \color{blue}{\frac{x}{\frac{t - z}{y - z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.45

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 5 \cdot 10^{-283}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.29%
Cost	1243

\[\begin{array}{l} \mathbf{if}\;y \leq -290 \lor \neg \left(y \leq -3.8 \cdot 10^{-21}\right) \land \left(y \leq -8.8 \cdot 10^{-64} \lor \neg \left(y \leq 5.5 \cdot 10^{-57}\right) \land \left(y \leq 3.6 \cdot 10^{-25} \lor \neg \left(y \leq 1.2 \cdot 10^{+21}\right)\right)\right):\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \end{array} \]

Alternative 2
Error	27.21%
Cost	1240

\[\begin{array}{l} t_1 := y \cdot \frac{x}{t - z}\\ t_2 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-296}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \end{array} \]

Alternative 3
Error	27.24%
Cost	1240

\[\begin{array}{l} t_1 := y \cdot \frac{x}{t - z}\\ t_2 := \frac{x}{\frac{t}{y - z}}\\ \mathbf{if}\;z \leq -7 \cdot 10^{-58}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-296}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \end{array} \]

Alternative 4
Error	27.31%
Cost	1240

\[\begin{array}{l} t_1 := y \cdot \frac{x}{t - z}\\ t_2 := \frac{x}{\frac{t}{y - z}}\\ \mathbf{if}\;z \leq -7 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-300}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \end{array} \]

Alternative 5
Error	27.28%
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ t_2 := \frac{x}{\frac{t - z}{y}}\\ \mathbf{if}\;y \leq -26000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-25}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	41.76%
Cost	1176

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+203}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{+180}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+62}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+70}:\\ \;\;\;\;\frac{-z}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	41.93%
Cost	1176

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+203}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{+180}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{+70}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.45 \cdot 10^{-68}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{-z}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	26.72%
Cost	1108

\[\begin{array}{l} t_1 := y \cdot \frac{x}{t - z}\\ t_2 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{-61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-302}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	40.91%
Cost	912

\[\begin{array}{l} t_1 := \frac{-z}{\frac{t}{x}}\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+62}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5.3 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	40.91%
Cost	912

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+66}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-68}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 10^{+70}:\\ \;\;\;\;\frac{-z}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	28.92%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-129} \lor \neg \left(z \leq 8.6 \cdot 10^{-76}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \end{array} \]

Alternative 12
Error	58.72%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-121}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	41.84%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 25000000000:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	40.86%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.32 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	3.52%
Cost	576

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

Alternative 16
Error	61.92%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(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)))