Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 7.4 → 1.3

Time: 12.0s

Precision: binary64

Cost: 1608

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \end{array} \]

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 -2e+293)
     (* (/ x z) (/ 2.0 (- y t)))
     (if (<= t_1 2e+224)
       (/ x (/ (* z (- y t)) 2.0))
       (/ (/ 2.0 z) (/ (- y t) x))))))

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+293) {
		tmp = (x / z) * (2.0 / (y - t));
	} else if (t_1 <= 2e+224) {
		tmp = x / ((z * (y - t)) / 2.0);
	} else {
		tmp = (2.0 / z) / ((y - t) / x);
	}
	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+293)) then
        tmp = (x / z) * (2.0d0 / (y - t))
    else if (t_1 <= 2d+224) then
        tmp = x / ((z * (y - t)) / 2.0d0)
    else
        tmp = (2.0d0 / z) / ((y - t) / x)
    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+293) {
		tmp = (x / z) * (2.0 / (y - t));
	} else if (t_1 <= 2e+224) {
		tmp = x / ((z * (y - t)) / 2.0);
	} else {
		tmp = (2.0 / z) / ((y - t) / x);
	}
	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+293:
		tmp = (x / z) * (2.0 / (y - t))
	elif t_1 <= 2e+224:
		tmp = x / ((z * (y - t)) / 2.0)
	else:
		tmp = (2.0 / z) / ((y - t) / x)
	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+293)
		tmp = Float64(Float64(x / z) * Float64(2.0 / Float64(y - t)));
	elseif (t_1 <= 2e+224)
		tmp = Float64(x / Float64(Float64(z * Float64(y - t)) / 2.0));
	else
		tmp = Float64(Float64(2.0 / z) / Float64(Float64(y - t) / x));
	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+293)
		tmp = (x / z) * (2.0 / (y - t));
	elseif (t_1 <= 2e+224)
		tmp = x / ((z * (y - t)) / 2.0);
	else
		tmp = (2.0 / z) / ((y - t) / x);
	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, -2e+293], N[(N[(x / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+224], N[(x / N[(N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.4
Target	2.3
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -1.9999999999999998e293

   1. Initial program 20.7

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

   2. Simplified 20.7

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

      [Start] 20.7	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-/l* [=>] 20.7	\[ \color{blue}{\frac{x}{\frac{y \cdot z - t \cdot z}{2}}} \]
      distribute-rgt-out-- [=>] 20.7	\[ \frac{x}{\frac{\color{blue}{z \cdot \left(y - t\right)}}{2}} \]

   3. Applied egg-rr 0.1

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

   if -1.9999999999999998e293 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1.99999999999999994e224

   1. Initial program 1.8

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

   2. Simplified 1.8

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

      [Start] 1.8	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-/l* [=>] 1.8	\[ \color{blue}{\frac{x}{\frac{y \cdot z - t \cdot z}{2}}} \]
      distribute-rgt-out-- [=>] 1.8	\[ \frac{x}{\frac{\color{blue}{z \cdot \left(y - t\right)}}{2}} \]

   if 1.99999999999999994e224 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 19.9

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

   2. Simplified 0.3

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

      [Start] 19.9	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      *-commutative [=>] 19.9	\[ \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 13.3	\[ \frac{2 \cdot x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      times-frac [=>] 0.3	\[ \color{blue}{\frac{2}{z} \cdot \frac{x}{y - t}} \]
      associate-*r/ [=>] 0.2	\[ \color{blue}{\frac{\frac{2}{z} \cdot x}{y - t}} \]
      associate-/l* [=>] 0.3	\[ \color{blue}{\frac{\frac{2}{z}}{\frac{y - t}{x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.8
Cost	976

\[\begin{array}{l} t_1 := x \cdot \frac{2}{y \cdot z}\\ t_2 := x \cdot \frac{-2}{z \cdot t}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.4 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 1.48 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	18.5
Cost	976

\[\begin{array}{l} t_1 := x \cdot \frac{2}{y \cdot z}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]

Alternative 3
Error	18.4
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]

Alternative 4
Error	18.4
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-72}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;t \leq -8.4 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{z \cdot t}{x}}\\ \end{array} \]

Alternative 5
Error	18.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{z \cdot t}{x}}\\ \end{array} \]

Alternative 6
Error	18.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq -8.4 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{z \cdot t}{x}}\\ \end{array} \]

Alternative 7
Error	6.0
Cost	973

\[\begin{array}{l} t_1 := \frac{2}{y - t}\\ \mathbf{if}\;t \leq -9 \cdot 10^{+252}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-106} \lor \neg \left(t \leq 4.8 \cdot 10^{-160}\right):\\ \;\;\;\;\frac{x}{z} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t_1}{z}\\ \end{array} \]

Alternative 8
Error	6.1
Cost	973

\[\begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{+253}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{+79} \lor \neg \left(t \leq 1.45 \cdot 10^{-163}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 9
Error	18.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{+63} \lor \neg \left(t \leq 1.6 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 10
Error	6.3
Cost	576

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

Alternative 11
Error	6.0
Cost	576

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

Alternative 12
Error	31.5
Cost	448

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

Reproduce

herbie shell --seed 2023016 
(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))))