Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Accuracy: 89.0% → 98.9%

Time: 10.6s

Precision: binary64

Cost: 2641

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+279}:\\ \;\;\;\;\frac{2}{\frac{z}{x} \cdot \left(y - t\right)}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-250} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq 10^{+108}:\\ \;\;\;\;\frac{2 \cdot x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]

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+279)
     (/ 2.0 (* (/ z x) (- y t)))
     (if (or (<= t_1 -5e-250) (and (not (<= t_1 0.0)) (<= t_1 1e+108)))
       (/ (* 2.0 x) (* z (- y t)))
       (* 2.0 (/ (/ x 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+279) {
		tmp = 2.0 / ((z / x) * (y - t));
	} else if ((t_1 <= -5e-250) || (!(t_1 <= 0.0) && (t_1 <= 1e+108))) {
		tmp = (2.0 * x) / (z * (y - t));
	} else {
		tmp = 2.0 * ((x / 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+279)) then
        tmp = 2.0d0 / ((z / x) * (y - t))
    else if ((t_1 <= (-5d-250)) .or. (.not. (t_1 <= 0.0d0)) .and. (t_1 <= 1d+108)) then
        tmp = (2.0d0 * x) / (z * (y - t))
    else
        tmp = 2.0d0 * ((x / 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+279) {
		tmp = 2.0 / ((z / x) * (y - t));
	} else if ((t_1 <= -5e-250) || (!(t_1 <= 0.0) && (t_1 <= 1e+108))) {
		tmp = (2.0 * x) / (z * (y - t));
	} else {
		tmp = 2.0 * ((x / 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+279:
		tmp = 2.0 / ((z / x) * (y - t))
	elif (t_1 <= -5e-250) or (not (t_1 <= 0.0) and (t_1 <= 1e+108)):
		tmp = (2.0 * x) / (z * (y - t))
	else:
		tmp = 2.0 * ((x / 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+279)
		tmp = Float64(2.0 / Float64(Float64(z / x) * Float64(y - t)));
	elseif ((t_1 <= -5e-250) || (!(t_1 <= 0.0) && (t_1 <= 1e+108)))
		tmp = Float64(Float64(2.0 * x) / Float64(z * Float64(y - t)));
	else
		tmp = Float64(2.0 * Float64(Float64(x / 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+279)
		tmp = 2.0 / ((z / x) * (y - t));
	elseif ((t_1 <= -5e-250) || (~((t_1 <= 0.0)) && (t_1 <= 1e+108)))
		tmp = (2.0 * x) / (z * (y - t));
	else
		tmp = 2.0 * ((x / 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+279], N[(2.0 / N[(N[(z / x), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, -5e-250], And[N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision], LessEqual[t$95$1, 1e+108]]], N[(N[(2.0 * x), $MachinePrecision] / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	89.0%
Target	96.7%
Herbie	98.9%

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -5.0000000000000002e279

   1. Initial program 73.1%

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

   2. Simplified 99.8%

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

      [Start] 73.1	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 73.1	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      times-frac [=>] 99.8	\[ \color{blue}{\frac{x}{z} \cdot \frac{2}{y - t}} \]

   3. Applied egg-rr 98.9%

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

      [Start] 99.8	\[ \frac{x}{z} \cdot \frac{2}{y - t} \]
      clear-num [=>] 99.8	\[ \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{2}{y - t} \]
      frac-times [=>] 98.9	\[ \color{blue}{\frac{1 \cdot 2}{\frac{z}{x} \cdot \left(y - t\right)}} \]
      metadata-eval [=>] 98.9	\[ \frac{\color{blue}{2}}{\frac{z}{x} \cdot \left(y - t\right)} \]

   if -5.0000000000000002e279 < (-.f64 (*.f64 y z) (*.f64 t z)) < -5.00000000000000027e-250 or 0.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1e108

   1. Initial program 99.6%

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

   2. Simplified 99.6%

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

      [Start] 99.6	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 99.6	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   if -5.00000000000000027e-250 < (-.f64 (*.f64 y z) (*.f64 t z)) < 0.0 or 1e108 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 75.6%

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

   2. Simplified 97.8%

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

      [Start] 75.6	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-*l/ [<=] 75.6	\[ \color{blue}{\frac{x}{y \cdot z - t \cdot z} \cdot 2} \]
      *-commutative [=>] 75.6	\[ \color{blue}{2 \cdot \frac{x}{y \cdot z - t \cdot z}} \]
      distribute-rgt-out-- [=>] 81.3	\[ 2 \cdot \frac{x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 97.8	\[ 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y - t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -5 \cdot 10^{+279}:\\ \;\;\;\;\frac{2}{\frac{z}{x} \cdot \left(y - t\right)}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq -5 \cdot 10^{-250} \lor \neg \left(y \cdot z - z \cdot t \leq 0\right) \land y \cdot z - z \cdot t \leq 10^{+108}:\\ \;\;\;\;\frac{2 \cdot x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	71.6%
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-202}:\\ \;\;\;\;\frac{-2}{\frac{z \cdot t}{x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \end{array} \]

Alternative 2
Accuracy	71.5%
Cost	976

\[\begin{array}{l} t_1 := \frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-202}:\\ \;\;\;\;\frac{-2}{\frac{z \cdot t}{x}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \end{array} \]

Alternative 3
Accuracy	71.7%
Cost	976

\[\begin{array}{l} t_1 := \frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{if}\;y \leq -8.6 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-202}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \end{array} \]

Alternative 4
Accuracy	95.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+99} \lor \neg \left(z \leq 2 \cdot 10^{-146}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \end{array} \]

Alternative 5
Accuracy	72.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-116} \lor \neg \left(y \leq 9.6 \cdot 10^{-30}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \end{array} \]

Alternative 6
Accuracy	71.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-116} \lor \neg \left(y \leq 9.5 \cdot 10^{-30}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]

Alternative 7
Accuracy	72.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \end{array} \]

Alternative 8
Accuracy	90.7%
Cost	576

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

Alternative 9
Accuracy	51.3%
Cost	448

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

Alternative 10
Accuracy	51.3%
Cost	448

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

Reproduce

herbie shell --seed 2023147 
(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))))