Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.7 → 1.2

Time: 13.3s

Precision: binary64

Cost: 1736

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+208}:\\ \;\;\;\;\frac{x \cdot 2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array} \]

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 -5e+277)
     (/ (/ x (- y t)) (* z 0.5))
     (if (<= t_1 5e+208) (/ (* x 2.0) t_1) (/ (/ (* x 2.0) z) (- y t))))))

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 <= -5e+277) {
		tmp = (x / (y - t)) / (z * 0.5);
	} else if (t_1 <= 5e+208) {
		tmp = (x * 2.0) / t_1;
	} else {
		tmp = ((x * 2.0) / z) / (y - t);
	}
	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 <= (-5d+277)) then
        tmp = (x / (y - t)) / (z * 0.5d0)
    else if (t_1 <= 5d+208) then
        tmp = (x * 2.0d0) / t_1
    else
        tmp = ((x * 2.0d0) / z) / (y - t)
    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 <= -5e+277) {
		tmp = (x / (y - t)) / (z * 0.5);
	} else if (t_1 <= 5e+208) {
		tmp = (x * 2.0) / t_1;
	} else {
		tmp = ((x * 2.0) / z) / (y - t);
	}
	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 <= -5e+277:
		tmp = (x / (y - t)) / (z * 0.5)
	elif t_1 <= 5e+208:
		tmp = (x * 2.0) / t_1
	else:
		tmp = ((x * 2.0) / z) / (y - t)
	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 <= -5e+277)
		tmp = Float64(Float64(x / Float64(y - t)) / Float64(z * 0.5));
	elseif (t_1 <= 5e+208)
		tmp = Float64(Float64(x * 2.0) / t_1);
	else
		tmp = Float64(Float64(Float64(x * 2.0) / z) / Float64(y - t));
	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 <= -5e+277)
		tmp = (x / (y - t)) / (z * 0.5);
	elseif (t_1 <= 5e+208)
		tmp = (x * 2.0) / t_1;
	else
		tmp = ((x * 2.0) / z) / (y - t);
	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, -5e+277], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+208], N[(N[(x * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.7
Target	2.1
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -4.99999999999999982e277

   1. Initial program 16.8

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

   2. Simplified 0.1

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

      [Start] 16.8	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      *-commutative [=>] 16.8	\[ \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 16.8	\[ \frac{2 \cdot x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      times-frac [=>] 0.1	\[ \color{blue}{\frac{2}{z} \cdot \frac{x}{y - t}} \]
      associate-*r/ [=>] 0.2	\[ \color{blue}{\frac{\frac{2}{z} \cdot x}{y - t}} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{\frac{2}{z}}{\frac{y - t}{x}}} \]

   3. Applied egg-rr 1.0

      \[\leadsto \color{blue}{{\left(\frac{y - t}{x} \cdot \left(z \cdot 0.5\right)\right)}^{-1}} \]

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{y - t}}{z \cdot 0.5}} \]
      Proof

      [Start] 1.0	\[ {\left(\frac{y - t}{x} \cdot \left(z \cdot 0.5\right)\right)}^{-1} \]
      unpow-1 [=>] 1.0	\[ \color{blue}{\frac{1}{\frac{y - t}{x} \cdot \left(z \cdot 0.5\right)}} \]
      associate-/r* [=>] 0.2	\[ \color{blue}{\frac{\frac{1}{\frac{y - t}{x}}}{z \cdot 0.5}} \]
      *-lft-identity [<=] 0.2	\[ \frac{\frac{1}{\color{blue}{1 \cdot \frac{y - t}{x}}}}{z \cdot 0.5} \]
      associate-*r/ [=>] 0.2	\[ \frac{\frac{1}{\color{blue}{\frac{1 \cdot \left(y - t\right)}{x}}}}{z \cdot 0.5} \]
      associate-*l/ [<=] 0.2	\[ \frac{\frac{1}{\color{blue}{\frac{1}{x} \cdot \left(y - t\right)}}}{z \cdot 0.5} \]
      associate-/r* [=>] 0.1	\[ \frac{\color{blue}{\frac{\frac{1}{\frac{1}{x}}}{y - t}}}{z \cdot 0.5} \]
      associate-/r/ [=>] 0.1	\[ \frac{\frac{\color{blue}{\frac{1}{1} \cdot x}}{y - t}}{z \cdot 0.5} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\color{blue}{1} \cdot x}{y - t}}{z \cdot 0.5} \]
      associate-*r/ [<=] 0.1	\[ \frac{\color{blue}{1 \cdot \frac{x}{y - t}}}{z \cdot 0.5} \]
      *-lft-identity [=>] 0.1	\[ \frac{\color{blue}{\frac{x}{y - t}}}{z \cdot 0.5} \]

   if -4.99999999999999982e277 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.0000000000000004e208

   1. Initial program 1.6

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

   if 5.0000000000000004e208 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 18.1

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

   2. Simplified 0.6

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

      [Start] 18.1	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 13.0	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 0.6	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 5 \cdot 10^{+208}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-247}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 2
Error	17.2
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-249}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 3
Error	17.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-247}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 4
Error	17.4
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-247}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-64}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{y \cdot z}{x}}\\ \end{array} \]

Alternative 5
Error	17.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq -8.4 \cdot 10^{-247}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]

Alternative 6
Error	17.4
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]

Alternative 7
Error	17.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.45 \cdot 10^{-247}:\\ \;\;\;\;\frac{\frac{x}{z}}{t \cdot -0.5}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-58}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]

Alternative 8
Error	2.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{+117} \lor \neg \left(z \leq 8.5 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 9
Error	2.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \end{array} \]

Alternative 10
Error	17.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+16} \lor \neg \left(y \leq 3.8 \cdot 10^{-57}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]

Alternative 11
Error	17.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+15} \lor \neg \left(y \leq 4.6 \cdot 10^{-58}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]

Alternative 12
Error	17.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 13
Error	5.5
Cost	576

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

Alternative 14
Error	5.7
Cost	576

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

Alternative 15
Error	31.7
Cost	448

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

Reproduce

herbie shell --seed 2023060 
(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))))