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

Average Accuracy: 81.8% → 96.2%

Time: 13.9s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-192}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;t \leq -8.4 \cdot 10^{-294}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.2e-192)
   (* x (/ (- z y) (- z t)))
   (if (<= t -8.4e-294) (/ (* x (- y z)) (- 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 (t <= -1.2e-192) {
		tmp = x * ((z - y) / (z - t));
	} else if (t <= -8.4e-294) {
		tmp = (x * (y - z)) / (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 (t <= (-1.2d-192)) then
        tmp = x * ((z - y) / (z - t))
    else if (t <= (-8.4d-294)) then
        tmp = (x * (y - z)) / (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 (t <= -1.2e-192) {
		tmp = x * ((z - y) / (z - t));
	} else if (t <= -8.4e-294) {
		tmp = (x * (y - z)) / (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 t <= -1.2e-192:
		tmp = x * ((z - y) / (z - t))
	elif t <= -8.4e-294:
		tmp = (x * (y - z)) / (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 (t <= -1.2e-192)
		tmp = Float64(x * Float64(Float64(z - y) / Float64(z - t)));
	elseif (t <= -8.4e-294)
		tmp = Float64(Float64(x * Float64(y - z)) / 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 (t <= -1.2e-192)
		tmp = x * ((z - y) / (z - t));
	elseif (t <= -8.4e-294)
		tmp = (x * (y - z)) / (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[t, -1.2e-192], N[(x * N[(N[(z - y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -8.4e-294], N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(t - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	81.8%
Target	97.0%
Herbie	96.2%

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

Derivation

1. Split input into 3 regimes

2. if t < -1.2e-192

   1. Initial program 81.6%

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

   2. Simplified 97.3%

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

      [Start] 81.6	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-*r/ [<=] 97.3	\[ \color{blue}{x \cdot \frac{y - z}{t - z}} \]
      sub-neg [=>] 97.3	\[ x \cdot \frac{\color{blue}{y + \left(-z\right)}}{t - z} \]
      +-commutative [=>] 97.3	\[ x \cdot \frac{\color{blue}{\left(-z\right) + y}}{t - z} \]
      neg-sub0 [=>] 97.3	\[ x \cdot \frac{\color{blue}{\left(0 - z\right)} + y}{t - z} \]
      associate-+l- [=>] 97.3	\[ x \cdot \frac{\color{blue}{0 - \left(z - y\right)}}{t - z} \]
      sub0-neg [=>] 97.3	\[ x \cdot \frac{\color{blue}{-\left(z - y\right)}}{t - z} \]
      neg-mul-1 [=>] 97.3	\[ x \cdot \frac{\color{blue}{-1 \cdot \left(z - y\right)}}{t - z} \]
      sub-neg [=>] 97.3	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{t + \left(-z\right)}} \]
      +-commutative [=>] 97.3	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(-z\right) + t}} \]
      neg-sub0 [=>] 97.3	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{\left(0 - z\right)} + t} \]
      associate-+l- [=>] 97.3	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{0 - \left(z - t\right)}} \]
      sub0-neg [=>] 97.3	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-\left(z - t\right)}} \]
      neg-mul-1 [=>] 97.3	\[ x \cdot \frac{-1 \cdot \left(z - y\right)}{\color{blue}{-1 \cdot \left(z - t\right)}} \]
      times-frac [=>] 97.3	\[ x \cdot \color{blue}{\left(\frac{-1}{-1} \cdot \frac{z - y}{z - t}\right)} \]
      metadata-eval [=>] 97.3	\[ x \cdot \left(\color{blue}{1} \cdot \frac{z - y}{z - t}\right) \]
      *-lft-identity [=>] 97.3	\[ x \cdot \color{blue}{\frac{z - y}{z - t}} \]

   if -1.2e-192 < t < -8.39999999999999937e-294

   1. Initial program 84.5%

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

   if -8.39999999999999937e-294 < t

   1. Initial program 81.6%

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

   2. Simplified 96.9%

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

      [Start] 81.6	\[ \frac{x \cdot \left(y - z\right)}{t - z} \]
      associate-/l* [=>] 96.9	\[ \color{blue}{\frac{x}{\frac{t - z}{y - z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-192}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;t \leq -8.4 \cdot 10^{-294}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	58.2%
Cost	1045

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+32}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-85}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+73} \lor \neg \left(z \leq 2.2 \cdot 10^{+101}\right) \land z \leq 8.2 \cdot 10^{+161}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	58.2%
Cost	1045

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-86}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+73} \lor \neg \left(z \leq 1.3 \cdot 10^{+101}\right) \land z \leq 1.2 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{\frac{t}{-z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	96.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-132} \lor \neg \left(z \leq 3 \cdot 10^{-226}\right):\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 4
Accuracy	96.8%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-132}:\\ \;\;\;\;x \cdot \frac{z - y}{z - t}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-223}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array} \]

Alternative 5
Accuracy	70.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{-55} \lor \neg \left(z \leq 1.2 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \end{array} \]

Alternative 6
Accuracy	74.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-24} \lor \neg \left(z \leq 7.5 \cdot 10^{-72}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{t - z}\\ \end{array} \]

Alternative 7
Accuracy	74.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-11} \lor \neg \left(z \leq 7.2 \cdot 10^{+45}\right):\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 8
Accuracy	74.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-10} \lor \neg \left(z \leq 2.6 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x}{\frac{z - t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 9
Accuracy	59.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+31}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	60.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+33}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	96.9%
Cost	576

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

Alternative 12
Accuracy	38.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023130 
(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)))