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

Average Accuracy: 82.0% → 97.2%

Time: 14.1s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{elif}\;t_1 \leq 10^{+302}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{\frac{z - t}{x}}\\ \end{array} \]

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 -1e-8)
     (* (- y z) (/ x (- t z)))
     (if (<= t_1 1e+302) 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 <= -1e-8) {
		tmp = (y - z) * (x / (t - z));
	} else if (t_1 <= 1e+302) {
		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 <= (-1d-8)) then
        tmp = (y - z) * (x / (t - z))
    else if (t_1 <= 1d+302) 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 <= -1e-8) {
		tmp = (y - z) * (x / (t - z));
	} else if (t_1 <= 1e+302) {
		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 <= -1e-8:
		tmp = (y - z) * (x / (t - z))
	elif t_1 <= 1e+302:
		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 <= -1e-8)
		tmp = Float64(Float64(y - z) * Float64(x / Float64(t - z)));
	elseif (t_1 <= 1e+302)
		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 <= -1e-8)
		tmp = (y - z) * (x / (t - z));
	elseif (t_1 <= 1e+302)
		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, -1e-8], N[(N[(y - z), $MachinePrecision] * N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+302], t$95$1, N[(N[(z - y), $MachinePrecision] / N[(N[(z - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	82.0%
Target	96.6%
Herbie	97.2%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -1e-8

   1. Initial program 63.8%

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

   2. Simplified 95.8%

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

      [Start] 63.8	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-*l/ [<=] 95.8	\[ \color{blue}{\frac{x}{t - z} \cdot \left(y - z\right)} \]

   if -1e-8 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 1.0000000000000001e302

   1. Initial program 97.4%

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

   if 1.0000000000000001e302 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 1.4%

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

   2. Simplified 99.6%

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

      [Start] 1.4	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-*r/ [<=] 99.6	\[ \color{blue}{x \cdot \frac{y - z}{t - z}} \]
      sub-neg [=>] 99.6	\[ x \cdot \frac{\color{blue}{y + \left(-z\right)}}{t - z} \]
      +-commutative [=>] 99.6	\[ x \cdot \frac{\color{blue}{\left(-z\right) + y}}{t - z} \]
      neg-sub0 [=>] 99.6	\[ x \cdot \frac{\color{blue}{\left(0 - z\right)} + y}{t - z} \]
      associate-+l- [=>] 99.6	\[ x \cdot \frac{\color{blue}{0 - \left(z - y\right)}}{t - z} \]
      sub0-neg [=>] 99.6	\[ x \cdot \frac{\color{blue}{-\left(z - y\right)}}{t - z} \]
      neg-mul-1 [=>] 99.6	\[ x \cdot \frac{\color{blue}{-1 \cdot \left(z - y\right)}}{t - z} \]
      sub-neg [=>] 99.6	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{t + \left(-z\right)}} \]
      +-commutative [=>] 99.6	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(-z\right) + t}} \]
      neg-sub0 [=>] 99.6	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(0 - z\right)} + t} \]
      associate-+l- [=>] 99.6	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{0 - \left(z - t\right)}} \]
      sub0-neg [=>] 99.6	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-\left(z - t\right)}} \]
      neg-mul-1 [=>] 99.6	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-1 \cdot \left(z - t\right)}} \]
      times-frac [=>] 99.6	\[ x \cdot \color{blue}{\left(\frac{-1}{-1} \cdot \frac{z - y}{z - t}\right)} \]
      metadata-eval [=>] 99.6	\[ x \cdot \left(\color{blue}{1} \cdot \frac{z - y}{z - t}\right) \]
      *-lft-identity [=>] 99.6	\[ x \cdot \color{blue}{\frac{z - y}{z - t}} \]

   3. Taylor expanded in x around 0 1.4%

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

   4. Simplified 99.1%

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

      [Start] 1.4	\[ \frac{\left(z - y\right) \cdot x}{z - t} \]
      associate-/l* [=>] 99.1	\[ \color{blue}{\frac{z - y}{\frac{z - t}{x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq 10^{+302}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{\frac{z - t}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	74.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-143}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+45}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \end{array} \]

Alternative 2
Accuracy	96.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-239} \lor \neg \left(z \leq 5.7 \cdot 10^{-294}\right):\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \end{array} \]

Alternative 3
Accuracy	96.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-240}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-286}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

Alternative 4
Accuracy	96.7%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-142}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-287}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

Alternative 5
Accuracy	71.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-111} \lor \neg \left(z \leq 1.75 \cdot 10^{-84}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \end{array} \]

Alternative 6
Accuracy	75.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-17} \lor \neg \left(z \leq 4 \cdot 10^{-53}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 7
Accuracy	73.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+72} \lor \neg \left(y \leq 2.5 \cdot 10^{+135}\right):\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \end{array} \]

Alternative 8
Accuracy	73.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+72} \lor \neg \left(y \leq 1.16 \cdot 10^{+135}\right):\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \end{array} \]

Alternative 9
Accuracy	60.3%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	61.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	38.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(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)))