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

Average Error: 11.6 → 0.6

Time: 12.5s

Precision: binary64

Cost: 3152

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ t_2 := x \cdot \frac{y - z}{t - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 1.2 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))) (t_2 (* x (/ (- y z) (- t z)))))
   (if (<= t_1 (- INFINITY))
     (/ x (/ (- t z) (- y z)))
     (if (<= t_1 -2e-298)
       t_1
       (if (<= t_1 0.0) t_2 (if (<= t_1 1.2e+183) t_1 t_2))))))

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 t_2 = x * ((y - z) / (t - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / ((t - z) / (y - z));
	} else if (t_1 <= -2e-298) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 1.2e+183) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 = (x * (y - z)) / (t - z);
	double t_2 = x * ((y - z) / (t - z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = x / ((t - z) / (y - z));
	} else if (t_1 <= -2e-298) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 1.2e+183) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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)
	t_2 = x * ((y - z) / (t - z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = x / ((t - z) / (y - z))
	elif t_1 <= -2e-298:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 1.2e+183:
		tmp = t_1
	else:
		tmp = t_2
	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))
	t_2 = Float64(x * Float64(Float64(y - z) / Float64(t - z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / Float64(Float64(t - z) / Float64(y - z)));
	elseif (t_1 <= -2e-298)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 1.2e+183)
		tmp = t_1;
	else
		tmp = t_2;
	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);
	t_2 = x * ((y - z) / (t - z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = x / ((t - z) / (y - z));
	elseif (t_1 <= -2e-298)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 1.2e+183)
		tmp = t_1;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(x * N[(N[(y - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / N[(N[(t - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-298], t$95$1, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 1.2e+183], t$95$1, t$95$2]]]]]]

Target

Original	11.6
Target	2.3
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{t - z}{y - z}}} \]
      Proof
      (/.f64 x (/.f64 (-.f64 t z) (-.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))): 73 points increase in error, 38 points decrease in error

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -1.99999999999999982e-298 or 0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.2000000000000001e183

   1. Initial program 0.3

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

   if -1.99999999999999982e-298 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 0.0 or 1.2000000000000001e183 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 26.8

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

   2. Simplified 1.4

      \[\leadsto \color{blue}{x \cdot \frac{y - z}{t - z}} \]
      Proof
      (*.f64 x (/.f64 (-.f64 y z) (-.f64 t z))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))): 75 points increase in error, 33 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -2 \cdot 10^{-298}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 0:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 1.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.9
Cost	1240

\[\begin{array}{l} t_1 := \frac{x \cdot y}{t - z}\\ t_2 := \frac{x}{1 - \frac{t}{z}}\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{z - y}{z}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-264}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-143}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	17.6
Cost	1240

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{z - y}{z}\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-102}:\\ \;\;\;\;\frac{z - y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-260}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-142}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{t}{z}}\\ \end{array} \]

Alternative 3
Error	16.7
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{1 - \frac{t}{z}}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{z - y}{z}\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-27}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.8
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{1 - \frac{t}{z}}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{z - y}{z}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{\frac{t - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	27.0
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -520000:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-96}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	27.1
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -255000:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-96}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	27.0
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -450000:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	27.1
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+127}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.04 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	2.5
Cost	840

\[\begin{array}{l} t_1 := x \cdot \frac{y - z}{t - z}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.02 \cdot 10^{-277}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	21.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.0086:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	18.4
Cost	712

\[\begin{array}{l} t_1 := x \cdot \frac{z - y}{z}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	16.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \frac{z - y}{z}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-29}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{t}{z}}\\ \end{array} \]

Alternative 13
Error	26.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-96}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	25.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	39.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022330 
(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)))