Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.8 → 0.7

Time: 9.3s

Precision: binary64

Cost: 2768

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ t_2 := \frac{2 \cdot x}{t_1}\\ t_3 := \frac{2}{z} \cdot \frac{x}{y - t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+212}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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)))
        (t_2 (/ (* 2.0 x) t_1))
        (t_3 (* (/ 2.0 z) (/ x (- y t)))))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 -1e-113)
       t_2
       (if (<= t_1 0.0)
         (* 2.0 (/ (/ x z) (- y t)))
         (if (<= t_1 2e+212) t_2 t_3))))))

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 t_2 = (2.0 * x) / t_1;
	double t_3 = (2.0 / z) * (x / (y - t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= -1e-113) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else if (t_1 <= 2e+212) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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 t_2 = (2.0 * x) / t_1;
	double t_3 = (2.0 / z) * (x / (y - t));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_1 <= -1e-113) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else if (t_1 <= 2e+212) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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)
	t_2 = (2.0 * x) / t_1
	t_3 = (2.0 / z) * (x / (y - t))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_3
	elif t_1 <= -1e-113:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = 2.0 * ((x / z) / (y - t))
	elif t_1 <= 2e+212:
		tmp = t_2
	else:
		tmp = t_3
	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))
	t_2 = Float64(Float64(2.0 * x) / t_1)
	t_3 = Float64(Float64(2.0 / z) * Float64(x / Float64(y - t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= -1e-113)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 * Float64(Float64(x / z) / Float64(y - t)));
	elseif (t_1 <= 2e+212)
		tmp = t_2;
	else
		tmp = t_3;
	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);
	t_2 = (2.0 * x) / t_1;
	t_3 = (2.0 / z) * (x / (y - t));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_3;
	elseif (t_1 <= -1e-113)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = 2.0 * ((x / z) / (y - t));
	elseif (t_1 <= 2e+212)
		tmp = t_2;
	else
		tmp = t_3;
	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]}, Block[{t$95$2 = N[(N[(2.0 * x), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 / z), $MachinePrecision] * N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, -1e-113], t$95$2, If[LessEqual[t$95$1, 0.0], N[(2.0 * N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+212], t$95$2, t$95$3]]]]]]]

Target

Original	6.8
Target	2.2
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -inf.0 or 1.9999999999999998e212 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 19.2

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

   2. Applied egg-rr 0.3

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

   if -inf.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < -9.99999999999999979e-114 or 0.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1.9999999999999998e212

   1. Initial program 0.3

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

   if -9.99999999999999979e-114 < (-.f64 (*.f64 y z) (*.f64 t z)) < 0.0

   1. Initial program 12.6

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

   2. Taylor expanded in x around 0 12.6

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

   3. Simplified 4.9

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y - t}} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 x z) (-.f64 y t))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 z (-.f64 y t))))): 42 points increase in error, 45 points decrease in error
      (*.f64 2 (/.f64 x (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 y z) (*.f64 t z))))): 2 points increase in error, 1 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq -1 \cdot 10^{-113}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z - z \cdot t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 0:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 2 \cdot 10^{+212}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.2
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{2 \cdot x}{y}}{z}\\ t_2 := \frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{if}\;t \leq -1.3379303326768917 \cdot 10^{-24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.4129038487417507 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.7436377980119867 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.613293916519135 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	17.9
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{2 \cdot x}{y}}{z}\\ t_2 := \frac{x \cdot \frac{-2}{t}}{z}\\ \mathbf{if}\;t \leq -1.3379303326768917 \cdot 10^{-24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.4129038487417507 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.7436377980119867 \cdot 10^{+39}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq 1.613293916519135 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	18.1
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{2 \cdot x}{y}}{z}\\ \mathbf{if}\;t \leq -1.3379303326768917 \cdot 10^{-24}:\\ \;\;\;\;\frac{x \cdot \frac{-2}{t}}{z}\\ \mathbf{elif}\;t \leq 1.4129038487417507 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.7436377980119867 \cdot 10^{+39}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq 1.613293916519135 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]

Alternative 4
Error	2.4
Cost	840

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{if}\;z \leq -6.17232459007547 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.03794323293498731:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	2.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -6.17232459007547 \cdot 10^{+58}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;z \leq 0.03794323293498731:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]

Alternative 6
Error	6.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1184495152674846 \cdot 10^{+184}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot x}{y}}{z}\\ \end{array} \]

Alternative 7
Error	30.0
Cost	448

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

Reproduce

herbie shell --seed 2022316 
(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))))