Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 7.1 → 0.7

Time: 8.2s

Precision: binary64

Cost: 2640

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ t_2 := 2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-152}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-243}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))) (t_2 (* 2.0 (/ (/ x z) (- y t)))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -1e-152)
       (/ x (/ (* z (- y t)) 2.0))
       (if (<= t_1 2e-243)
         t_2
         (if (<= t_1 5e+140) (* x (/ (/ 2.0 (- y t)) z)) t_2))))))

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 / z) / (y - t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -1e-152) {
		tmp = x / ((z * (y - t)) / 2.0);
	} else if (t_1 <= 2e-243) {
		tmp = t_2;
	} else if (t_1 <= 5e+140) {
		tmp = x * ((2.0 / (y - t)) / z);
	} else {
		tmp = t_2;
	}
	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 / z) / (y - t));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= -1e-152) {
		tmp = x / ((z * (y - t)) / 2.0);
	} else if (t_1 <= 2e-243) {
		tmp = t_2;
	} else if (t_1 <= 5e+140) {
		tmp = x * ((2.0 / (y - t)) / z);
	} else {
		tmp = t_2;
	}
	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 / z) / (y - t))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= -1e-152:
		tmp = x / ((z * (y - t)) / 2.0)
	elif t_1 <= 2e-243:
		tmp = t_2
	elif t_1 <= 5e+140:
		tmp = x * ((2.0 / (y - t)) / z)
	else:
		tmp = t_2
	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(2.0 * Float64(Float64(x / z) / Float64(y - t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -1e-152)
		tmp = Float64(x / Float64(Float64(z * Float64(y - t)) / 2.0));
	elseif (t_1 <= 2e-243)
		tmp = t_2;
	elseif (t_1 <= 5e+140)
		tmp = Float64(x * Float64(Float64(2.0 / Float64(y - t)) / z));
	else
		tmp = t_2;
	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 / z) / (y - t));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= -1e-152)
		tmp = x / ((z * (y - t)) / 2.0);
	elseif (t_1 <= 2e-243)
		tmp = t_2;
	elseif (t_1 <= 5e+140)
		tmp = x * ((2.0 / (y - t)) / z);
	else
		tmp = t_2;
	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[(2.0 * N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -1e-152], N[(x / N[(N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-243], t$95$2, If[LessEqual[t$95$1, 5e+140], N[(x * N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Target

Original	7.1
Target	2.4
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 -1.00000000000000007e-152 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1.99999999999999999e-243 or 5.00000000000000008e140 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 16.9

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

   2. Simplified 1.3

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y - t}} \]
      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))))): 60 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))))): 10 points increase in error, 1 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 x) (-.f64 (*.f64 y z) (*.f64 t z)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x 2)) (-.f64 (*.f64 y z) (*.f64 t z))): 0 points increase in error, 0 points decrease in error

   if -inf.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < -1.00000000000000007e-152

   1. Initial program 0.2

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}} \]
      Proof
      (/.f64 x (/.f64 (*.f64 z (-.f64 y t)) 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 y z) (*.f64 t z))) 2)): 10 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z)))): 0 points increase in error, 0 points decrease in error

   if 1.99999999999999999e-243 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.00000000000000008e140

   1. Initial program 0.3

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{x \cdot \frac{\frac{2}{y - t}}{z}} \]
      Proof
      (*.f64 x (/.f64 (/.f64 2 (-.f64 y t)) z)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 z (-.f64 y t))))): 24 points increase in error, 19 points decrease in error
      (*.f64 x (/.f64 2 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 y z) (*.f64 t z))))): 10 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z)))): 16 points increase in error, 30 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -\infty:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq -1 \cdot 10^{-152}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 2 \cdot 10^{-243}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 5 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.8
Cost	976

\[\begin{array}{l} t_1 := \frac{2}{y \cdot \frac{z}{x}}\\ t_2 := -2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{if}\;t \leq -480:\\ \;\;\;\;\frac{\frac{x}{t}}{z} \cdot -2\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	2.9
Cost	840

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	28.2
Cost	712

\[\begin{array}{l} t_1 := -2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	18.1
Cost	712

\[\begin{array}{l} t_1 := x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-68}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	17.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -0.000108:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-70}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]

Alternative 6
Error	17.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -510:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]

Alternative 7
Error	17.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -480:\\ \;\;\;\;\frac{\frac{x}{t}}{z} \cdot -2\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]

Alternative 8
Error	5.8
Cost	576

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

Alternative 9
Error	29.7
Cost	448

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

Reproduce

herbie shell --seed 2022330 
(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))))