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

Average Error: 11.6 → 2.0

Time: 12.4s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \end{array} \]

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 -5e+69)
     (* (- y z) (/ x (- t z)))
     (if (<= t_1 -4e-152) 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 <= -5e+69) {
		tmp = (y - z) * (x / (t - z));
	} else if (t_1 <= -4e-152) {
		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 <= (-5d+69)) then
        tmp = (y - z) * (x / (t - z))
    else if (t_1 <= (-4d-152)) 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 <= -5e+69) {
		tmp = (y - z) * (x / (t - z));
	} else if (t_1 <= -4e-152) {
		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 <= -5e+69:
		tmp = (y - z) * (x / (t - z))
	elif t_1 <= -4e-152:
		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 <= -5e+69)
		tmp = Float64(Float64(y - z) * Float64(x / Float64(t - z)));
	elseif (t_1 <= -4e-152)
		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 <= -5e+69)
		tmp = (y - z) * (x / (t - z));
	elseif (t_1 <= -4e-152)
		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, -5e+69], N[(N[(y - z), $MachinePrecision] * N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-152], t$95$1, N[(x * N[(N[(z - y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	11.6
Target	2.0
Herbie	2.0

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -5.00000000000000036e69

   1. Initial program 29.6

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

   2. Simplified 3.7

      \[\leadsto \color{blue}{\frac{x}{t - z} \cdot \left(y - z\right)} \]
      Proof
      (*.f64 (/.f64 x (-.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))): 2 points increase in error, 0 points decrease in error

   if -5.00000000000000036e69 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < -4.00000000000000026e-152

   1. Initial program 0.3

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

   if -4.00000000000000026e-152 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 9.6

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

   2. Simplified 1.9

      \[\leadsto \color{blue}{x \cdot \frac{z - y}{z - t}} \]
      Proof
      (*.f64 (/.f64 x (-.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))): 2 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \leq -4 \cdot 10^{-152}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.3
Cost	1109

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+189}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq -0.01:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-92}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-92} \lor \neg \left(z \leq 2.3 \cdot 10^{-46}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \end{array} \]

Alternative 2
Error	18.4
Cost	977

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;z \leq -8 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-87}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-93} \lor \neg \left(z \leq 3.1 \cdot 10^{-71}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \end{array} \]

Alternative 3
Error	17.1
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{+189}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq -0.0019:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.05 \cdot 10^{-117}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.2
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+189}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq -0.1:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-43}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \end{array} \]

Alternative 5
Error	17.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+189}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq -0.0046:\\ \;\;\;\;\frac{x}{\frac{z}{z - y}}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \end{array} \]

Alternative 6
Error	17.1
Cost	845

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+189}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq -0.145 \lor \neg \left(z \leq 2.6 \cdot 10^{-41}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 7
Error	16.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.005 \lor \neg \left(z \leq 2.4 \cdot 10^{-71}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 8
Error	37.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	25.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -0.057:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	25.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -0.46:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	1.9
Cost	576

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

Alternative 12
Error	40.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022343 
(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)))