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

Average Error: 18.1% → 2.42%

Time: 15.0s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+225}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \end{array} \]

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 0.0)
     (/ x (/ (- t z) (- y z)))
     (if (<= t_1 2e+225) 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 <= 0.0) {
		tmp = x / ((t - z) / (y - z));
	} else if (t_1 <= 2e+225) {
		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 <= 0.0d0) then
        tmp = x / ((t - z) / (y - z))
    else if (t_1 <= 2d+225) 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 <= 0.0) {
		tmp = x / ((t - z) / (y - z));
	} else if (t_1 <= 2e+225) {
		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 <= 0.0:
		tmp = x / ((t - z) / (y - z))
	elif t_1 <= 2e+225:
		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 <= 0.0)
		tmp = Float64(x / Float64(Float64(t - z) / Float64(y - z)));
	elseif (t_1 <= 2e+225)
		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 <= 0.0)
		tmp = x / ((t - z) / (y - z));
	elseif (t_1 <= 2e+225)
		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, 0.0], N[(x / N[(N[(t - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+225], t$95$1, N[(x * N[(N[(z - y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	18.1%
Target	3.41%
Herbie	2.42%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 0.0

   1. Initial program 17.43

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

   2. Simplified 3.42

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

      [Start] 17.43	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-/l* [=>] 3.42	\[ \color{blue}{\frac{x}{\frac{t - z}{y - z}}} \]

   if 0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.99999999999999986e225

   1. Initial program 0.52

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

   if 1.99999999999999986e225 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 80.74

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

   2. Simplified 3.56

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

      [Start] 80.74	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-*r/ [<=] 3.56	\[ \color{blue}{x \cdot \frac{y - z}{t - z}} \]
      sub-neg [=>] 3.56	\[ x \cdot \frac{\color{blue}{y + \left(-z\right)}}{t - z} \]
      +-commutative [=>] 3.56	\[ x \cdot \frac{\color{blue}{\left(-z\right) + y}}{t - z} \]
      neg-sub0 [=>] 3.56	\[ x \cdot \frac{\color{blue}{\left(0 - z\right)} + y}{t - z} \]
      associate-+l- [=>] 3.56	\[ x \cdot \frac{\color{blue}{0 - \left(z - y\right)}}{t - z} \]
      sub0-neg [=>] 3.56	\[ x \cdot \frac{\color{blue}{-\left(z - y\right)}}{t - z} \]
      neg-mul-1 [=>] 3.56	\[ x \cdot \frac{\color{blue}{-1 \cdot \left(z - y\right)}}{t - z} \]
      sub-neg [=>] 3.56	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{t + \left(-z\right)}} \]
      +-commutative [=>] 3.56	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(-z\right) + t}} \]
      neg-sub0 [=>] 3.56	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(0 - z\right)} + t} \]
      associate-+l- [=>] 3.56	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{0 - \left(z - t\right)}} \]
      sub0-neg [=>] 3.56	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-\left(z - t\right)}} \]
      neg-mul-1 [=>] 3.56	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-1 \cdot \left(z - t\right)}} \]
      times-frac [=>] 3.56	\[ x \cdot \color{blue}{\left(\frac{-1}{-1} \cdot \frac{z - y}{z - t}\right)} \]
      metadata-eval [=>] 3.56	\[ x \cdot \left(\color{blue}{1} \cdot \frac{z - y}{z - t}\right) \]
      *-lft-identity [=>] 3.56	\[ x \cdot \color{blue}{\frac{z - y}{z - t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.42

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 0:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 2 \cdot 10^{+225}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	28.18%
Cost	1373

\[\begin{array}{l} t_1 := \frac{x}{1 - \frac{t}{z}}\\ t_2 := x \cdot \frac{y}{t - z}\\ t_3 := x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -17000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -63000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+60} \lor \neg \left(y \leq 1.05 \cdot 10^{+111}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	28.22%
Cost	1373

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{z}\right)\\ t_2 := x \cdot \frac{y}{t - z}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3550000000000:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{elif}\;y \leq -50000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{1 - \frac{t}{z}}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+59} \lor \neg \left(y \leq 3.8 \cdot 10^{+114}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	29.7%
Cost	1372

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{z}\right)\\ t_2 := \frac{x \cdot y}{t - z}\\ \mathbf{if}\;y \leq -8.6 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -64000000000000:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{elif}\;y \leq -13500:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.45 \cdot 10^{-78}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{1 - \frac{t}{z}}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \end{array} \]

Alternative 4
Error	41.58%
Cost	1112

\[\begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+172}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-97}:\\ \;\;\;\;\left(-x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-295}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	41.54%
Cost	1112

\[\begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+172}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;z \leq -2.95 \cdot 10^{-291}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	40.03%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.38 \cdot 10^{-98}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-290}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	40.12%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.62 \cdot 10^{-99}:\\ \;\;\;\;\left(-x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-288}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	26.16%
Cost	844

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-254}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	4.86%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{-131} \lor \neg \left(z \leq 3.8 \cdot 10^{-302}\right):\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \end{array} \]

Alternative 10
Error	3.38%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-15} \lor \neg \left(z \leq 4.9 \cdot 10^{-256}\right):\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 11
Error	3.43%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-251}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

Alternative 12
Error	30.83%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{-95} \lor \neg \left(z \leq 3.6 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \end{array} \]

Alternative 13
Error	26.78%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-24} \lor \neg \left(z \leq 4.2 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \end{array} \]

Alternative 14
Error	58.77%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-136}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-32}:\\ \;\;\;\;z \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	39.33%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	39.36%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	61.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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)))