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

Average Error: 11.4 → 1.5

Time: 15.8s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;x \cdot \frac{1}{\frac{t - z}{y - z}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+280}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{\frac{z - t}{x}}\\ \end{array} \]

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 (/ 1.0 (/ (- t z) (- y z))))
     (if (<= t_1 2e+280) t_1 (/ (- z y) (/ (- z t) x))))))

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 * (1.0 / ((t - z) / (y - z)));
	} else if (t_1 <= 2e+280) {
		tmp = t_1;
	} else {
		tmp = (z - y) / ((z - t) / x);
	}
	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 * (1.0d0 / ((t - z) / (y - z)))
    else if (t_1 <= 2d+280) then
        tmp = t_1
    else
        tmp = (z - y) / ((z - t) / x)
    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 * (1.0 / ((t - z) / (y - z)));
	} else if (t_1 <= 2e+280) {
		tmp = t_1;
	} else {
		tmp = (z - y) / ((z - t) / x);
	}
	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 * (1.0 / ((t - z) / (y - z)))
	elif t_1 <= 2e+280:
		tmp = t_1
	else:
		tmp = (z - y) / ((z - t) / x)
	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(1.0 / Float64(Float64(t - z) / Float64(y - z))));
	elseif (t_1 <= 2e+280)
		tmp = t_1;
	else
		tmp = Float64(Float64(z - y) / Float64(Float64(z - t) / x));
	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 * (1.0 / ((t - z) / (y - z)));
	elseif (t_1 <= 2e+280)
		tmp = t_1;
	else
		tmp = (z - y) / ((z - t) / x);
	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[(1.0 / N[(N[(t - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+280], t$95$1, N[(N[(z - y), $MachinePrecision] / N[(N[(z - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	11.4
Target	2.2
Herbie	1.5

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.1

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

   2. Applied egg-rr 2.3

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{t - z}{y - z}}} \]

   if 0.0 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 2.0000000000000001e280

   1. Initial program 0.2

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

   if 2.0000000000000001e280 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 59.7

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

   2. Simplified 1.6

      \[\leadsto \color{blue}{\left(z - y\right) \cdot \frac{x}{z - t}} \]
      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)))): 21 points increase in error, 83 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))): 86 points increase in error, 28 points decrease in error

   3. Applied egg-rr 1.5

      \[\leadsto \color{blue}{\frac{z - y}{\frac{z - t}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 0:\\ \;\;\;\;x \cdot \frac{1}{\frac{t - z}{y - z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{\frac{z - t}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	1864

\[\begin{array}{l} t_1 := \frac{z - y}{\frac{z - t}{x}}\\ t_2 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+237}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	23.5
Cost	1504

\[\begin{array}{l} t_1 := \frac{x}{\frac{-z}{y}}\\ t_2 := \frac{y}{\frac{t - z}{x}}\\ \mathbf{if}\;z \leq -8.990729740474396 \cdot 10^{+231}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.9445018300927004 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.237861212104173 \cdot 10^{+145}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.7295603141929982 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.378221650469589 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 10^{-205}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 6.062142329906094 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	23.7
Cost	1504

\[\begin{array}{l} t_1 := \frac{x}{\frac{t - z}{y}}\\ \mathbf{if}\;z \leq -8.990729740474396 \cdot 10^{+231}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.9445018300927004 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.237861212104173 \cdot 10^{+145}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.7295603141929982 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.378221650469589 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-187}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;z \leq 1.0030590481304201 \cdot 10^{-23}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 4
Error	23.8
Cost	1504

\[\begin{array}{l} t_1 := \frac{x}{\frac{t - z}{y}}\\ \mathbf{if}\;z \leq -8.990729740474396 \cdot 10^{+231}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.9445018300927004 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.237861212104173 \cdot 10^{+145}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.7295603141929982 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.378221650469589 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-160}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 10^{-187}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \mathbf{elif}\;z \leq 1.0030590481304201 \cdot 10^{-23}:\\ \;\;\;\;\frac{x \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 5
Error	22.2
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{\frac{t - z}{y}}\\ \mathbf{if}\;z \leq -8.990729740474396 \cdot 10^{+231}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.9445018300927004 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.237861212104173 \cdot 10^{+145}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.7295603141929982 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.378221650469589 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.581877445164262 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	24.5
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{\frac{t - z}{y}}\\ \mathbf{if}\;z \leq -8.990729740474396 \cdot 10^{+231}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.9445018300927004 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.237861212104173 \cdot 10^{+145}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.7295603141929982 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.378221650469589 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 7
Error	27.0
Cost	1112

\[\begin{array}{l} t_1 := \frac{x}{\frac{-z}{y}}\\ \mathbf{if}\;z \leq -8.990729740474396 \cdot 10^{+231}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.9445018300927004 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.237861212104173 \cdot 10^{+145}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.7295603141929982 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1023.7872750665111:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	17.8
Cost	976

\[\begin{array}{l} t_1 := x \cdot \frac{y}{t - z}\\ t_2 := \frac{x}{\frac{z}{z - y}}\\ \mathbf{if}\;z \leq -1023.7872750665111:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;z \leq 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	16.2
Cost	976

\[\begin{array}{l} t_1 := x \cdot \frac{y}{t - z}\\ \mathbf{if}\;z \leq -1023.7872750665111:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;z \leq 1.0030590481304201 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \end{array} \]

Alternative 10
Error	10.6
Cost	972

\[\begin{array}{l} t_1 := \frac{z - y}{\frac{z - t}{x}}\\ \mathbf{if}\;z \leq -2.401931129916439 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-120}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	10.2
Cost	972

\[\begin{array}{l} t_1 := \left(z - y\right) \cdot \frac{x}{z - t}\\ \mathbf{if}\;z \leq -2.401931129916439 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-120}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	17.8
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{\frac{z}{z - y}}\\ \mathbf{if}\;z \leq -1023.7872750665111:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-100}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	25.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1023.7872750665111:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{y}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	25.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1023.7872750665111:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	25.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1023.7872750665111:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	39.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022308 
(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)))