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

Average Accuracy: 82.5% → 96.7%

Time: 15.4s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-77}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-254}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.1e-77)
   (* x (/ (- y z) (- t z)))
   (if (<= z -3e-254) (* (- y z) (/ x (- t z))) (/ x (/ (- t z) (- y z))))))

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 tmp;
	if (z <= -3.1e-77) {
		tmp = x * ((y - z) / (t - z));
	} else if (z <= -3e-254) {
		tmp = (y - z) * (x / (t - z));
	} else {
		tmp = x / ((t - z) / (y - z));
	}
	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) :: tmp
    if (z <= (-3.1d-77)) then
        tmp = x * ((y - z) / (t - z))
    else if (z <= (-3d-254)) then
        tmp = (y - z) * (x / (t - z))
    else
        tmp = x / ((t - z) / (y - z))
    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 tmp;
	if (z <= -3.1e-77) {
		tmp = x * ((y - z) / (t - z));
	} else if (z <= -3e-254) {
		tmp = (y - z) * (x / (t - z));
	} else {
		tmp = x / ((t - z) / (y - z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	tmp = 0
	if z <= -3.1e-77:
		tmp = x * ((y - z) / (t - z))
	elif z <= -3e-254:
		tmp = (y - z) * (x / (t - z))
	else:
		tmp = x / ((t - z) / (y - z))
	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)
	tmp = 0.0
	if (z <= -3.1e-77)
		tmp = Float64(x * Float64(Float64(y - z) / Float64(t - z)));
	elseif (z <= -3e-254)
		tmp = Float64(Float64(y - z) * Float64(x / Float64(t - z)));
	else
		tmp = Float64(x / Float64(Float64(t - z) / Float64(y - z)));
	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)
	tmp = 0.0;
	if (z <= -3.1e-77)
		tmp = x * ((y - z) / (t - z));
	elseif (z <= -3e-254)
		tmp = (y - z) * (x / (t - z));
	else
		tmp = x / ((t - z) / (y - z));
	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_] := If[LessEqual[z, -3.1e-77], N[(x * N[(N[(y - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3e-254], N[(N[(y - z), $MachinePrecision] * N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(t - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	82.5%
Target	96.9%
Herbie	96.7%

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

Derivation

1. Split input into 3 regimes

2. if z < -3.10000000000000008e-77

   1. Initial program 77.7%

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

   2. Simplified 99.4%

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

      [Start] 77.7	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-*r/ [<=] 99.4	\[ \color{blue}{x \cdot \frac{y - z}{t - z}} \]

   if -3.10000000000000008e-77 < z < -3.00000000000000012e-254

   1. Initial program 90.9%

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

   2. Simplified 91.7%

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

      [Start] 90.9	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-*l/ [<=] 91.7	\[ \color{blue}{\frac{x}{t - z} \cdot \left(y - z\right)} \]

   if -3.00000000000000012e-254 < z

   1. Initial program 83.3%

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

   2. Simplified 96.4%

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

      [Start] 83.3	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-/l* [=>] 96.4	\[ \color{blue}{\frac{x}{\frac{t - z}{y - z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-77}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-254}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.6%
Cost	1240

\[\begin{array}{l} t_1 := \frac{y}{\frac{t - z}{x}}\\ t_2 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;z \leq -106:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-185}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	73.6%
Cost	1240

\[\begin{array}{l} t_1 := \frac{y}{\frac{t - z}{x}}\\ t_2 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;z \leq -11.2:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{-123}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-64}:\\ \;\;\;\;\frac{y - z}{\frac{t}{x}}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	65.4%
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -2900000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+38}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	73.7%
Cost	844

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;z \leq -490:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.38 \cdot 10^{-157}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	73.8%
Cost	844

\[\begin{array}{l} t_1 := x \cdot \frac{z}{z - t}\\ \mathbf{if}\;z \leq -20:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-162}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{\frac{t - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	96.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-78} \lor \neg \left(z \leq -3.1 \cdot 10^{-254}\right):\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 7
Accuracy	65.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3100000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+39}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	59.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -3000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	60.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -46000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	60.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -14200:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	97.0%
Cost	576

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

Alternative 12
Accuracy	37.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(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)))