Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.9 → 1.4

Time: 10.8s

Precision: binary64

Cost: 1737

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+165} \lor \neg \left(t_1 \leq 5 \cdot 10^{+209}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{t_1}\\ \end{array} \]

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 -1e+165) (not (<= t_1 5e+209)))
     (* (/ x z) (/ 2.0 (- 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 <= -1e+165) || !(t_1 <= 5e+209)) {
		tmp = (x / z) * (2.0 / (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 <= (-1d+165)) .or. (.not. (t_1 <= 5d+209))) then
        tmp = (x / z) * (2.0d0 / (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 <= -1e+165) || !(t_1 <= 5e+209)) {
		tmp = (x / z) * (2.0 / (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 <= -1e+165) or not (t_1 <= 5e+209):
		tmp = (x / z) * (2.0 / (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 <= -1e+165) || !(t_1 <= 5e+209))
		tmp = Float64(Float64(x / z) * Float64(2.0 / 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 <= -1e+165) || ~((t_1 <= 5e+209)))
		tmp = (x / z) * (2.0 / (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, -1e+165], N[Not[LessEqual[t$95$1, 5e+209]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Target

Original	6.9
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)) < -9.99999999999999899e164 or 4.99999999999999964e209 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 14.4

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

   2. Simplified 11.8

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

      [Start] 14.4	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-/l* [=>] 14.4	\[ \color{blue}{\frac{x}{\frac{y \cdot z - t \cdot z}{2}}} \]
      distribute-rgt-out-- [=>] 11.8	\[ \frac{x}{\frac{\color{blue}{z \cdot \left(y - t\right)}}{2}} \]

   3. Applied egg-rr 0.7

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

   if -9.99999999999999899e164 < (-.f64 (*.f64 y z) (*.f64 t z)) < 4.99999999999999964e209

   1. Initial program 1.8

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -1 \cdot 10^{+165} \lor \neg \left(y \cdot z - z \cdot t \leq 5 \cdot 10^{+209}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z - z \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.0
Cost	977

\[\begin{array}{l} t_1 := \frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -0.00023:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-81} \lor \neg \left(y \leq 1.58 \cdot 10^{+87}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]

Alternative 2
Error	2.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-47} \lor \neg \left(z \leq 2.3 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \end{array} \]

Alternative 3
Error	2.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-53} \lor \neg \left(z \leq 2.8 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 4
Error	2.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+126} \lor \neg \left(z \leq 10^{-19}\right):\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 5
Error	17.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -250000000 \lor \neg \left(t \leq 82000000000000\right):\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 6
Error	17.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+27} \lor \neg \left(t \leq 1.2 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 7
Error	17.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{elif}\;t \leq 11000000000000:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]

Alternative 8
Error	17.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{elif}\;t \leq 4250000000000:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]

Alternative 9
Error	18.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 1850000000000:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]

Alternative 10
Error	6.0
Cost	576

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

Alternative 11
Error	32.0
Cost	448

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

Reproduce

herbie shell --seed 2023039 
(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))))