Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Accuracy: 89.1% → 97.2%

Time: 10.9s

Precision: binary64

Cost: 2120

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot 2}{y \cdot z - z \cdot t}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-323}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot \frac{z}{2}}\\ \mathbf{elif}\;t_1 \leq 20:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\ \end{array} \]

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 (/ (* x 2.0) (- (* y z) (* z t)))))
   (if (<= t_1 -2e-323)
     (/ x (* (- y t) (/ z 2.0)))
     (if (<= t_1 20.0)
       (* 2.0 (/ (/ x z) (- y t)))
       (/ (/ (* x 2.0) (- y t)) 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 = (x * 2.0) / ((y * z) - (z * t));
	double tmp;
	if (t_1 <= -2e-323) {
		tmp = x / ((y - t) * (z / 2.0));
	} else if (t_1 <= 20.0) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else {
		tmp = ((x * 2.0) / (y - t)) / 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 = (x * 2.0d0) / ((y * z) - (z * t))
    if (t_1 <= (-2d-323)) then
        tmp = x / ((y - t) * (z / 2.0d0))
    else if (t_1 <= 20.0d0) then
        tmp = 2.0d0 * ((x / z) / (y - t))
    else
        tmp = ((x * 2.0d0) / (y - t)) / 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 = (x * 2.0) / ((y * z) - (z * t));
	double tmp;
	if (t_1 <= -2e-323) {
		tmp = x / ((y - t) * (z / 2.0));
	} else if (t_1 <= 20.0) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else {
		tmp = ((x * 2.0) / (y - t)) / 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 = (x * 2.0) / ((y * z) - (z * t))
	tmp = 0
	if t_1 <= -2e-323:
		tmp = x / ((y - t) * (z / 2.0))
	elif t_1 <= 20.0:
		tmp = 2.0 * ((x / z) / (y - t))
	else:
		tmp = ((x * 2.0) / (y - t)) / 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(x * 2.0) / Float64(Float64(y * z) - Float64(z * t)))
	tmp = 0.0
	if (t_1 <= -2e-323)
		tmp = Float64(x / Float64(Float64(y - t) * Float64(z / 2.0)));
	elseif (t_1 <= 20.0)
		tmp = Float64(2.0 * Float64(Float64(x / z) / Float64(y - t)));
	else
		tmp = Float64(Float64(Float64(x * 2.0) / Float64(y - t)) / 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 = (x * 2.0) / ((y * z) - (z * t));
	tmp = 0.0;
	if (t_1 <= -2e-323)
		tmp = x / ((y - t) * (z / 2.0));
	elseif (t_1 <= 20.0)
		tmp = 2.0 * ((x / z) / (y - t));
	else
		tmp = ((x * 2.0) / (y - t)) / 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[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-323], N[(x / N[(N[(y - t), $MachinePrecision] * N[(z / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 20.0], N[(2.0 * N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	89.1%
Target	96.5%
Herbie	97.2%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z))) < -1.97626e-323

   1. Initial program 97.1%

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

   2. Simplified 97.2%

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

      [Start] 97.1	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 97.1	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 87.0	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]
      associate-/l* [=>] 87.0	\[ \frac{\color{blue}{\frac{x}{\frac{z}{2}}}}{y - t} \]
      associate-/l/ [=>] 97.2	\[ \color{blue}{\frac{x}{\left(y - t\right) \cdot \frac{z}{2}}} \]

   if -1.97626e-323 < (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z))) < 20

   1. Initial program 86.0%

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

   2. Simplified 98.0%

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

      [Start] 86.0	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-*l/ [<=] 86.0	\[ \color{blue}{\frac{x}{y \cdot z - t \cdot z} \cdot 2} \]
      *-commutative [=>] 86.0	\[ \color{blue}{2 \cdot \frac{x}{y \cdot z - t \cdot z}} \]
      distribute-rgt-out-- [=>] 86.0	\[ 2 \cdot \frac{x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 98.0	\[ 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y - t}} \]

   if 20 < (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 82.1%

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

   2. Simplified 91.2%

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

      [Start] 82.1	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-*r/ [<=] 81.6	\[ \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]
      distribute-rgt-out-- [=>] 91.2	\[ x \cdot \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Applied egg-rr 94.8%

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

      [Start] 91.2	\[ x \cdot \frac{2}{z \cdot \left(y - t\right)} \]
      associate-*r/ [=>] 91.6	\[ \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
      *-commutative [=>] 91.6	\[ \frac{x \cdot 2}{\color{blue}{\left(y - t\right) \cdot z}} \]
      associate-/r* [=>] 94.8	\[ \color{blue}{\frac{\frac{x \cdot 2}{y - t}}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -2 \cdot 10^{-323}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot \frac{z}{2}}\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq 20:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-92} \lor \neg \left(z \leq 7.2 \cdot 10^{+25}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \end{array} \]

Alternative 2
Accuracy	95.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+179} \lor \neg \left(z \leq 3.35 \cdot 10^{+24}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t - y}}{z}\\ \end{array} \]

Alternative 3
Accuracy	72.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{+19} \lor \neg \left(t \leq 2.4 \cdot 10^{-18}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \end{array} \]

Alternative 4
Accuracy	72.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+17} \lor \neg \left(t \leq 3.6 \cdot 10^{-26}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]

Alternative 5
Accuracy	72.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2.85 \cdot 10^{+19} \lor \neg \left(t \leq 1.85 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]

Alternative 6
Accuracy	90.6%
Cost	576

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

Alternative 7
Accuracy	50.4%
Cost	448

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

Alternative 8
Accuracy	50.4%
Cost	448

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

Reproduce

herbie shell --seed 2023151 
(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))))