Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 7.2 → 1.4

Time: 9.8s

Precision: binary64

Cost: 1737

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+165} \lor \neg \left(t_1 \leq 5 \cdot 10^{+193}\right):\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{t_1}\\ \end{array} \]

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 (or (<= t_1 -2e+165) (not (<= t_1 5e+193)))
     (/ (/ (* x 2.0) z) (- y t))
     (/ (* x 2.0) t_1))))

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 <= -2e+165) || !(t_1 <= 5e+193)) {
		tmp = ((x * 2.0) / z) / (y - t);
	} else {
		tmp = (x * 2.0) / t_1;
	}
	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 * 2.0d0) / ((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) :: t_1
    real(8) :: tmp
    t_1 = (y * z) - (z * t)
    if ((t_1 <= (-2d+165)) .or. (.not. (t_1 <= 5d+193))) then
        tmp = ((x * 2.0d0) / z) / (y - t)
    else
        tmp = (x * 2.0d0) / t_1
    end if
    code = tmp
end function

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 <= -2e+165) || !(t_1 <= 5e+193)) {
		tmp = ((x * 2.0) / z) / (y - t);
	} else {
		tmp = (x * 2.0) / t_1;
	}
	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 <= -2e+165) or not (t_1 <= 5e+193):
		tmp = ((x * 2.0) / z) / (y - t)
	else:
		tmp = (x * 2.0) / t_1
	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 <= -2e+165) || !(t_1 <= 5e+193))
		tmp = Float64(Float64(Float64(x * 2.0) / z) / Float64(y - t));
	else
		tmp = Float64(Float64(x * 2.0) / t_1);
	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 <= -2e+165) || ~((t_1 <= 5e+193)))
		tmp = ((x * 2.0) / z) / (y - t);
	else
		tmp = (x * 2.0) / t_1;
	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[Or[LessEqual[t$95$1, -2e+165], N[Not[LessEqual[t$95$1, 5e+193]], $MachinePrecision]], N[(N[(N[(x * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Target

Original	7.2
Target	2.2
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -1.9999999999999998e165 or 4.99999999999999972e193 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 14.4

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

   2. Simplified 0.6

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

      [Start] 14.4	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 11.7	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 0.6	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]

   if -1.9999999999999998e165 < (-.f64 (*.f64 y z) (*.f64 t z)) < 4.99999999999999972e193

   1. Initial program 2.0

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -2 \cdot 10^{+165} \lor \neg \left(y \cdot z - z \cdot t \leq 5 \cdot 10^{+193}\right):\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z - z \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.4
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-53} \lor \neg \left(y \leq 2.6 \cdot 10^{-143}\right) \land \left(y \leq 1.25 \cdot 10^{-25} \lor \neg \left(y \leq 1.9 \cdot 10^{+87}\right)\right):\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \end{array} \]

Alternative 2
Error	19.6
Cost	977

\[\begin{array}{l} t_1 := \frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-21} \lor \neg \left(y \leq 1.2 \cdot 10^{+101}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \end{array} \]

Alternative 3
Error	19.7
Cost	976

\[\begin{array}{l} t_1 := \frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+101}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	20.0
Cost	976

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+101}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{y}{\frac{x}{z}}}\\ \end{array} \]

Alternative 5
Error	20.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+101}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{y}{\frac{x}{z}}}\\ \end{array} \]

Alternative 6
Error	2.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+58} \lor \neg \left(z \leq 1.9 \cdot 10^{+83}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \end{array} \]

Alternative 7
Error	2.4
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+56} \lor \neg \left(z \leq 5.8 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 8
Error	2.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]

Alternative 9
Error	6.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+171}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{y}{\frac{x}{z}}}\\ \end{array} \]

Alternative 10
Error	31.2
Cost	448

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

Reproduce

herbie shell --seed 2023018 
(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))))