Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Accuracy: 89.3% → 97.5%

Time: 11.0s

Precision: binary64

Cost: 2892

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{x}{y - t}}{z \cdot 0.5}\\ t_2 := \frac{x \cdot 2}{y \cdot z - z \cdot t}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{elif}\;t_2 \leq 10^{+204}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ (/ x (- y t)) (* z 0.5)))
        (t_2 (/ (* x 2.0) (- (* y z) (* z t)))))
   (if (<= t_2 -1e+27)
     t_1
     (if (<= t_2 0.0)
       (/ (/ (* x 2.0) z) (- y t))
       (if (<= t_2 1e+204) (/ x (/ (* z (- y t)) 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 = (x / (y - t)) / (z * 0.5);
	double t_2 = (x * 2.0) / ((y * z) - (z * t));
	double tmp;
	if (t_2 <= -1e+27) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = ((x * 2.0) / z) / (y - t);
	} else if (t_2 <= 1e+204) {
		tmp = x / ((z * (y - t)) / 2.0);
	} else {
		tmp = 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) :: t_2
    real(8) :: tmp
    t_1 = (x / (y - t)) / (z * 0.5d0)
    t_2 = (x * 2.0d0) / ((y * z) - (z * t))
    if (t_2 <= (-1d+27)) then
        tmp = t_1
    else if (t_2 <= 0.0d0) then
        tmp = ((x * 2.0d0) / z) / (y - t)
    else if (t_2 <= 1d+204) then
        tmp = x / ((z * (y - t)) / 2.0d0)
    else
        tmp = 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 = (x / (y - t)) / (z * 0.5);
	double t_2 = (x * 2.0) / ((y * z) - (z * t));
	double tmp;
	if (t_2 <= -1e+27) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = ((x * 2.0) / z) / (y - t);
	} else if (t_2 <= 1e+204) {
		tmp = x / ((z * (y - t)) / 2.0);
	} else {
		tmp = 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 = (x / (y - t)) / (z * 0.5)
	t_2 = (x * 2.0) / ((y * z) - (z * t))
	tmp = 0
	if t_2 <= -1e+27:
		tmp = t_1
	elif t_2 <= 0.0:
		tmp = ((x * 2.0) / z) / (y - t)
	elif t_2 <= 1e+204:
		tmp = x / ((z * (y - t)) / 2.0)
	else:
		tmp = 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(x / Float64(y - t)) / Float64(z * 0.5))
	t_2 = Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(z * t)))
	tmp = 0.0
	if (t_2 <= -1e+27)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(x * 2.0) / z) / Float64(y - t));
	elseif (t_2 <= 1e+204)
		tmp = Float64(x / Float64(Float64(z * Float64(y - t)) / 2.0));
	else
		tmp = 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 = (x / (y - t)) / (z * 0.5);
	t_2 = (x * 2.0) / ((y * z) - (z * t));
	tmp = 0.0;
	if (t_2 <= -1e+27)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = ((x * 2.0) / z) / (y - t);
	elseif (t_2 <= 1e+204)
		tmp = x / ((z * (y - t)) / 2.0);
	else
		tmp = 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[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+27], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(x * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+204], N[(x / N[(N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Target

Original	89.3%
Target	96.6%
Herbie	97.5%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z))) < -1e27 or 9.99999999999999989e203 < (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 81.4%

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

   2. Simplified 93.6%

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

      [Start] 81.4	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      *-commutative [=>] 81.4	\[ \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 89.8	\[ \frac{2 \cdot x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      times-frac [=>] 93.2	\[ \color{blue}{\frac{2}{z} \cdot \frac{x}{y - t}} \]
      associate-*r/ [=>] 72.0	\[ \color{blue}{\frac{\frac{2}{z} \cdot x}{y - t}} \]
      associate-/l* [=>] 93.6	\[ \color{blue}{\frac{\frac{2}{z}}{\frac{y - t}{x}}} \]

   3. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{{\left(\frac{y - t}{x} \cdot \left(z \cdot 0.5\right)\right)}^{-1}} \]

   4. Simplified 93.4%

      \[\leadsto \color{blue}{\frac{\frac{x}{y - t}}{z \cdot 0.5}} \]
      Proof

      [Start] 93.6	\[ {\left(\frac{y - t}{x} \cdot \left(z \cdot 0.5\right)\right)}^{-1} \]
      unpow-1 [=>] 93.6	\[ \color{blue}{\frac{1}{\frac{y - t}{x} \cdot \left(z \cdot 0.5\right)}} \]
      associate-/r* [=>] 93.2	\[ \color{blue}{\frac{\frac{1}{\frac{y - t}{x}}}{z \cdot 0.5}} \]
      *-lft-identity [<=] 93.2	\[ \frac{\frac{1}{\color{blue}{1 \cdot \frac{y - t}{x}}}}{z \cdot 0.5} \]
      associate-*r/ [=>] 93.2	\[ \frac{\frac{1}{\color{blue}{\frac{1 \cdot \left(y - t\right)}{x}}}}{z \cdot 0.5} \]
      associate-*l/ [<=] 93.1	\[ \frac{\frac{1}{\color{blue}{\frac{1}{x} \cdot \left(y - t\right)}}}{z \cdot 0.5} \]
      associate-/r* [=>] 93.2	\[ \frac{\color{blue}{\frac{\frac{1}{\frac{1}{x}}}{y - t}}}{z \cdot 0.5} \]
      associate-/r/ [=>] 93.4	\[ \frac{\frac{\color{blue}{\frac{1}{1} \cdot x}}{y - t}}{z \cdot 0.5} \]
      metadata-eval [=>] 93.4	\[ \frac{\frac{\color{blue}{1} \cdot x}{y - t}}{z \cdot 0.5} \]
      associate-*r/ [<=] 93.4	\[ \frac{\color{blue}{1 \cdot \frac{x}{y - t}}}{z \cdot 0.5} \]
      *-lft-identity [=>] 93.4	\[ \frac{\color{blue}{\frac{x}{y - t}}}{z \cdot 0.5} \]

   if -1e27 < (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z))) < -0.0

   1. Initial program 86.6%

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

   2. Simplified 98.0%

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

      [Start] 86.6	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 86.6	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 98.0	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]

   if -0.0 < (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z))) < 9.99999999999999989e203

   1. Initial program 99.4%

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

   2. Simplified 99.3%

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

      [Start] 99.4	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-/l* [=>] 99.3	\[ \color{blue}{\frac{x}{\frac{y \cdot z - t \cdot z}{2}}} \]
      distribute-rgt-out-- [=>] 99.3	\[ \frac{x}{\frac{\color{blue}{z \cdot \left(y - t\right)}}{2}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq 0:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq 10^{+204}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+104} \lor \neg \left(x \leq 2.75 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 2
Accuracy	95.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+104} \lor \neg \left(x \leq 5.4 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 3
Accuracy	72.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -800000 \lor \neg \left(t \leq 1.8 \cdot 10^{-37}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \end{array} \]

Alternative 4
Accuracy	72.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -5500000 \lor \neg \left(t \leq 1.3 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	72.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -2150000000:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \end{array} \]

Alternative 6
Accuracy	73.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -21000000:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \end{array} \]

Alternative 7
Accuracy	72.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -17500000000:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{z}{\frac{x}{t}}}\\ \end{array} \]

Alternative 8
Accuracy	72.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -230000000:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{z}{\frac{x}{t}}}\\ \end{array} \]

Alternative 9
Accuracy	90.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+233}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 10
Accuracy	50.7%
Cost	448

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

Reproduce

herbie shell --seed 2023122 
(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))))