Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.6% → 99.1%

Time: 11.5s

Precision: binary64

Cost: 2642

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+198} \lor \neg \left(t_1 \leq -1 \cdot 10^{-118}\right) \land \left(t_1 \leq 0 \lor \neg \left(t_1 \leq 5 \cdot 10^{+268}\right)\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t - y}}{z}\\ \end{array} \]

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 -4e+198)
           (and (not (<= t_1 -1e-118))
                (or (<= t_1 0.0) (not (<= t_1 5e+268)))))
     (* 2.0 (/ (/ x z) (- y t)))
     (* x (/ (/ -2.0 (- t y)) z)))))

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 <= -4e+198) || (!(t_1 <= -1e-118) && ((t_1 <= 0.0) || !(t_1 <= 5e+268)))) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else {
		tmp = x * ((-2.0 / (t - y)) / z);
	}
	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 <= (-4d+198)) .or. (.not. (t_1 <= (-1d-118))) .and. (t_1 <= 0.0d0) .or. (.not. (t_1 <= 5d+268))) then
        tmp = 2.0d0 * ((x / z) / (y - t))
    else
        tmp = x * (((-2.0d0) / (t - y)) / z)
    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 <= -4e+198) || (!(t_1 <= -1e-118) && ((t_1 <= 0.0) || !(t_1 <= 5e+268)))) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else {
		tmp = x * ((-2.0 / (t - y)) / z);
	}
	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 <= -4e+198) or (not (t_1 <= -1e-118) and ((t_1 <= 0.0) or not (t_1 <= 5e+268))):
		tmp = 2.0 * ((x / z) / (y - t))
	else:
		tmp = x * ((-2.0 / (t - y)) / z)
	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 <= -4e+198) || (!(t_1 <= -1e-118) && ((t_1 <= 0.0) || !(t_1 <= 5e+268))))
		tmp = Float64(2.0 * Float64(Float64(x / z) / Float64(y - t)));
	else
		tmp = Float64(x * Float64(Float64(-2.0 / Float64(t - y)) / z));
	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 <= -4e+198) || (~((t_1 <= -1e-118)) && ((t_1 <= 0.0) || ~((t_1 <= 5e+268)))))
		tmp = 2.0 * ((x / z) / (y - t));
	else
		tmp = x * ((-2.0 / (t - y)) / z);
	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, -4e+198], And[N[Not[LessEqual[t$95$1, -1e-118]], $MachinePrecision], Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 5e+268]], $MachinePrecision]]]], N[(2.0 * N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(-2.0 / N[(t - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	89.6%
Target	96.9%
Herbie	99.1%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -4.00000000000000007e198 or -9.99999999999999985e-119 < (-.f64 (*.f64 y z) (*.f64 t z)) < -0.0 or 5.0000000000000002e268 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 76.4%

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

   2. Simplified 99.8%

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

      [Start] 76.4	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-*l/ [<=] 76.4	\[ \color{blue}{\frac{x}{y \cdot z - t \cdot z} \cdot 2} \]
      *-commutative [=>] 76.4	\[ \color{blue}{2 \cdot \frac{x}{y \cdot z - t \cdot z}} \]
      distribute-rgt-out-- [=>] 76.7	\[ 2 \cdot \frac{x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 99.8	\[ 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y - t}} \]

   if -4.00000000000000007e198 < (-.f64 (*.f64 y z) (*.f64 t z)) < -9.99999999999999985e-119 or -0.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.0000000000000002e268

   1. Initial program 99.5%

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

   2. Simplified 99.7%

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

      [Start] 99.5	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-*r/ [<=] 99.5	\[ \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]
      distribute-rgt-out-- [=>] 99.5	\[ x \cdot \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/l/ [<=] 99.7	\[ x \cdot \color{blue}{\frac{\frac{2}{y - t}}{z}} \]
      sub-neg [=>] 99.7	\[ x \cdot \frac{\frac{2}{\color{blue}{y + \left(-t\right)}}}{z} \]
      +-commutative [=>] 99.7	\[ x \cdot \frac{\frac{2}{\color{blue}{\left(-t\right) + y}}}{z} \]
      neg-sub0 [=>] 99.7	\[ x \cdot \frac{\frac{2}{\color{blue}{\left(0 - t\right)} + y}}{z} \]
      associate-+l- [=>] 99.7	\[ x \cdot \frac{\frac{2}{\color{blue}{0 - \left(t - y\right)}}}{z} \]
      sub0-neg [=>] 99.7	\[ x \cdot \frac{\frac{2}{\color{blue}{-\left(t - y\right)}}}{z} \]
      neg-mul-1 [=>] 99.7	\[ x \cdot \frac{\frac{2}{\color{blue}{-1 \cdot \left(t - y\right)}}}{z} \]
      associate-/r* [=>] 99.7	\[ x \cdot \frac{\color{blue}{\frac{\frac{2}{-1}}{t - y}}}{z} \]
      metadata-eval [=>] 99.7	\[ x \cdot \frac{\frac{\color{blue}{-2}}{t - y}}{z} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -4 \cdot 10^{+198} \lor \neg \left(y \cdot z - z \cdot t \leq -1 \cdot 10^{-118}\right) \land \left(y \cdot z - z \cdot t \leq 0 \lor \neg \left(y \cdot z - z \cdot t \leq 5 \cdot 10^{+268}\right)\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t - y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.2%
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	73.2%
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{\frac{t}{-2}}}{z}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	74.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+26} \lor \neg \left(t \leq 2.1 \cdot 10^{-20}\right):\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \end{array} \]

Alternative 4
Accuracy	73.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -1.82 \cdot 10^{+19} \lor \neg \left(t \leq 1.8 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \end{array} \]

Alternative 5
Accuracy	90.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{+120}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{t}{-2}}}{z}\\ \end{array} \]

Alternative 6
Accuracy	93.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+52}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]

Alternative 7
Accuracy	55.7%
Cost	448

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

Reproduce

herbie shell --seed 2023160 
(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))))