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

Average Error: 11.7 → 1.2

Time: 12.2s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_1 := \left(z - y\right) \cdot \frac{x}{z - t}\\ t_2 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- z y) (/ x (- z t)))) (t_2 (/ (* x (- y z)) (- t z))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+280) t_2 t_1))))

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 = (z - y) * (x / (z - t));
	double t_2 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+280) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 = (z - y) * (x / (z - t));
	double t_2 = (x * (y - z)) / (t - z);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 5e+280) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = (z - y) * (x / (z - t))
	t_2 = (x * (y - z)) / (t - z)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 5e+280:
		tmp = t_2
	else:
		tmp = t_1
	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(z - y) * Float64(x / Float64(z - t)))
	t_2 = Float64(Float64(x * Float64(y - z)) / Float64(t - z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+280)
		tmp = t_2;
	else
		tmp = t_1;
	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 = (z - y) * (x / (z - t));
	t_2 = (x * (y - z)) / (t - z);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 5e+280)
		tmp = t_2;
	else
		tmp = t_1;
	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[(z - y), $MachinePrecision] * N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+280], t$95$2, t$95$1]]]]

Target

Original	11.7
Target	2.3
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -inf.0 or 5.0000000000000002e280 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 61.9

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

   2. Simplified 0.6

      \[\leadsto \color{blue}{\left(z - y\right) \cdot \frac{x}{z - t}} \]
      Proof
      (*.f64 (-.f64 z y) (/.f64 x (-.f64 z t))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 x (-.f64 z t)) (-.f64 z y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 x (/.f64 (-.f64 z t) (-.f64 z y)))): 28 points increase in error, 66 points decrease in error
      (/.f64 x (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 z t) (-.f64 z y))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 z t) (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 z t)) (*.f64 -1 (-.f64 z y))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 z t))) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z t))) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) t)) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) t) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 z))) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 t z)) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 z y))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z y))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) y))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (Rewrite<= +-commutative_binary64 (+.f64 y (neg.f64 z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (-.f64 t z) (Rewrite<= sub-neg_binary64 (-.f64 y z)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))): 66 points increase in error, 30 points decrease in error

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 5.0000000000000002e280

   1. Initial program 1.3

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

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

Alternatives

Alternative 1
Error	18.1
Cost	1108

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{x}{t}\\ t_2 := x - \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;z \leq -5.250669456302135 \cdot 10^{+81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -6.963918086958537 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1974892950732293600:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.706242161549109 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.804321732432168 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{\frac{t - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	6.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.473910470434564 \cdot 10^{+110}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{elif}\;z \leq 4.43539790008324 \cdot 10^{+139}:\\ \;\;\;\;\frac{z - y}{\frac{z - t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	6.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -6.133462432847258 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{elif}\;z \leq 4.43539790008324 \cdot 10^{+139}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Error	20.3
Cost	712

\[\begin{array}{l} t_1 := x - \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;z \leq -1974892950732293600:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.5505147736117427 \cdot 10^{-75}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	18.4
Cost	712

\[\begin{array}{l} t_1 := x - \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;z \leq -5.250669456302135 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.804321732432168 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{\frac{t - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	16.1
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{\frac{z - t}{z}}\\ \mathbf{if}\;z \leq -2.6108307544432555 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.804321732432168 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{\frac{t - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	16.4
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{\frac{z - t}{z}}\\ \mathbf{if}\;z \leq -2.6108307544432555 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.804321732432168 \cdot 10^{-44}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	16.1
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{\frac{z - t}{z}}\\ \mathbf{if}\;z \leq -2.6108307544432555 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.804321732432168 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	26.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -5.250669456302135 \cdot 10^{+81}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.804321732432168 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	25.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -5.250669456302135 \cdot 10^{+81}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.804321732432168 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	39.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022317 
(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)))