Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Accuracy: 89.6% → 98.8%

Time: 12.0s

Precision: binary64

Cost: 2641

Math

\[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-57}:\\ \;\;\;\;\frac{2 \cdot x}{t_1}\\ \mathbf{elif}\;t_1 \leq 0 \lor \neg \left(t_1 \leq 4 \cdot 10^{+243}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) (* z t))))
   (if (<= t_1 (- INFINITY))
     (/ (/ 2.0 z) (/ (- y t) x))
     (if (<= t_1 -2e-57)
       (/ (* 2.0 x) t_1)
       (if (or (<= t_1 0.0) (not (<= t_1 4e+243)))
         (* 2.0 (/ (/ x z) (- y t)))
         (/ x (/ (* z (- y t)) 2.0)))))))

C

double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (2.0 / z) / ((y - t) / x);
	} else if (t_1 <= -2e-57) {
		tmp = (2.0 * x) / t_1;
	} else if ((t_1 <= 0.0) || !(t_1 <= 4e+243)) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else {
		tmp = x / ((z * (y - t)) / 2.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - (z * t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (2.0 / z) / ((y - t) / x);
	} else if (t_1 <= -2e-57) {
		tmp = (2.0 * x) / t_1;
	} else if ((t_1 <= 0.0) || !(t_1 <= 4e+243)) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else {
		tmp = x / ((z * (y - t)) / 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (x * 2.0) / ((y * z) - (t * z))

↓

def code(x, y, z, t):
	t_1 = (y * z) - (z * t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (2.0 / z) / ((y - t) / x)
	elif t_1 <= -2e-57:
		tmp = (2.0 * x) / t_1
	elif (t_1 <= 0.0) or not (t_1 <= 4e+243):
		tmp = 2.0 * ((x / z) / (y - t))
	else:
		tmp = x / ((z * (y - t)) / 2.0)
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
end

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(2.0 / z) / Float64(Float64(y - t) / x));
	elseif (t_1 <= -2e-57)
		tmp = Float64(Float64(2.0 * x) / t_1);
	elseif ((t_1 <= 0.0) || !(t_1 <= 4e+243))
		tmp = Float64(2.0 * Float64(Float64(x / z) / Float64(y - t)));
	else
		tmp = Float64(x / Float64(Float64(z * Float64(y - t)) / 2.0));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * 2.0) / ((y * z) - (t * z));
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) - (z * t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (2.0 / z) / ((y - t) / x);
	elseif (t_1 <= -2e-57)
		tmp = (2.0 * x) / t_1;
	elseif ((t_1 <= 0.0) || ~((t_1 <= 4e+243)))
		tmp = 2.0 * ((x / z) / (y - t));
	else
		tmp = x / ((z * (y - t)) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(2.0 / z), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-57], N[(N[(2.0 * x), $MachinePrecision] / t$95$1), $MachinePrecision], If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 4e+243]], $MachinePrecision]], N[(2.0 * N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	89.6%
Target	96.7%
Herbie	98.8%

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -inf.0

   1. Initial program 68.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{z}}{\frac{y - t}{x}}} \]
      Proof

      [Start] 68.2	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      *-commutative [=>] 68.2	\[ \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 68.2	\[ \frac{2 \cdot x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      times-frac [=>] 99.8	\[ \color{blue}{\frac{2}{z} \cdot \frac{x}{y - t}} \]
      associate-*r/ [=>] 99.8	\[ \color{blue}{\frac{\frac{2}{z} \cdot x}{y - t}} \]
      associate-/l* [=>] 99.8	\[ \color{blue}{\frac{\frac{2}{z}}{\frac{y - t}{x}}} \]

   if -inf.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < -1.99999999999999991e-57

   1. Initial program 99.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   if -1.99999999999999991e-57 < (-.f64 (*.f64 y z) (*.f64 t z)) < -0.0 or 4.0000000000000003e243 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 74.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Simplified 96.5%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y - t}} \]
      Proof

      [Start] 74.7	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      *-commutative [=>] 74.7	\[ \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]
      associate-*r/ [<=] 74.7	\[ \color{blue}{2 \cdot \frac{x}{y \cdot z - t \cdot z}} \]
      distribute-rgt-out-- [=>] 80.6	\[ 2 \cdot \frac{x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 96.5	\[ 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y - t}} \]

   if -0.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < 4.0000000000000003e243

   1. Initial program 99.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}} \]
      Proof

      [Start] 99.5	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-/l* [=>] 99.5	\[ \color{blue}{\frac{x}{\frac{y \cdot z - t \cdot z}{2}}} \]
      distribute-rgt-out-- [=>] 99.5	\[ \frac{x}{\frac{\color{blue}{z \cdot \left(y - t\right)}}{2}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq -2 \cdot 10^{-57}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z - z \cdot t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 0 \lor \neg \left(y \cdot z - z \cdot t \leq 4 \cdot 10^{+243}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	72.8%
Cost	976

\[\begin{array}{l} t_1 := \frac{-2}{t \cdot \frac{z}{x}}\\ t_2 := 2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+90}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	72.9%
Cost	976

\[\begin{array}{l} t_1 := \frac{-2}{t \cdot \frac{z}{x}}\\ t_2 := 2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\frac{y \cdot z}{x}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	73.0%
Cost	976

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\frac{y \cdot z}{x}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	73.1%
Cost	976

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{if}\;y \leq -7 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 950000000000:\\ \;\;\;\;\frac{\frac{-2}{t}}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+90}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+95}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	72.8%
Cost	976

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{x}{z}}{t \cdot -0.5}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	96.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+86} \lor \neg \left(z \leq 5.1 \cdot 10^{-48}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \end{array} \]

Alternative 7
Accuracy	96.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+86} \lor \neg \left(z \leq 5 \cdot 10^{-45}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \end{array} \]

Alternative 8
Accuracy	96.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+91} \lor \neg \left(z \leq 2 \cdot 10^{-49}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 9
Accuracy	72.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+33} \lor \neg \left(y \leq 3500000\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]

Alternative 10
Accuracy	72.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+32} \lor \neg \left(y \leq 6500000\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 11
Accuracy	89.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+125}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]

Alternative 12
Accuracy	50.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+82}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 13
Accuracy	50.6%
Cost	448

\[2 \cdot \frac{\frac{x}{y}}{z} \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))