Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.4% → 96.3%

Time: 14.1s

Precision: binary64

Cost: 2252

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-206}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{2}{y - t}}{\frac{z}{x}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+194}:\\ \;\;\;\;\frac{x \cdot 2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{2} \cdot \frac{y - t}{x}}\\ \end{array} \]

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-206)
     (* x (/ 2.0 (* z (- y t))))
     (if (<= t_1 5e-279)
       (/ (/ 2.0 (- y t)) (/ z x))
       (if (<= t_1 5e+194)
         (/ (* x 2.0) t_1)
         (/ 1.0 (* (/ z 2.0) (/ (- y t) x))))))))

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-206) {
		tmp = x * (2.0 / (z * (y - t)));
	} else if (t_1 <= 5e-279) {
		tmp = (2.0 / (y - t)) / (z / x);
	} else if (t_1 <= 5e+194) {
		tmp = (x * 2.0) / t_1;
	} else {
		tmp = 1.0 / ((z / 2.0) * ((y - t) / x));
	}
	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-206)) then
        tmp = x * (2.0d0 / (z * (y - t)))
    else if (t_1 <= 5d-279) then
        tmp = (2.0d0 / (y - t)) / (z / x)
    else if (t_1 <= 5d+194) then
        tmp = (x * 2.0d0) / t_1
    else
        tmp = 1.0d0 / ((z / 2.0d0) * ((y - t) / x))
    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-206) {
		tmp = x * (2.0 / (z * (y - t)));
	} else if (t_1 <= 5e-279) {
		tmp = (2.0 / (y - t)) / (z / x);
	} else if (t_1 <= 5e+194) {
		tmp = (x * 2.0) / t_1;
	} else {
		tmp = 1.0 / ((z / 2.0) * ((y - t) / x));
	}
	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-206:
		tmp = x * (2.0 / (z * (y - t)))
	elif t_1 <= 5e-279:
		tmp = (2.0 / (y - t)) / (z / x)
	elif t_1 <= 5e+194:
		tmp = (x * 2.0) / t_1
	else:
		tmp = 1.0 / ((z / 2.0) * ((y - t) / x))
	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-206)
		tmp = Float64(x * Float64(2.0 / Float64(z * Float64(y - t))));
	elseif (t_1 <= 5e-279)
		tmp = Float64(Float64(2.0 / Float64(y - t)) / Float64(z / x));
	elseif (t_1 <= 5e+194)
		tmp = Float64(Float64(x * 2.0) / t_1);
	else
		tmp = Float64(1.0 / Float64(Float64(z / 2.0) * Float64(Float64(y - t) / x)));
	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-206)
		tmp = x * (2.0 / (z * (y - t)));
	elseif (t_1 <= 5e-279)
		tmp = (2.0 / (y - t)) / (z / x);
	elseif (t_1 <= 5e+194)
		tmp = (x * 2.0) / t_1;
	else
		tmp = 1.0 / ((z / 2.0) * ((y - t) / x));
	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-206], N[(x * N[(2.0 / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-279], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+194], N[(N[(x * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(1.0 / N[(N[(z / 2.0), $MachinePrecision] * N[(N[(y - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	89.4%
Target	97.0%
Herbie	96.3%

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -5e-206

   1. Initial program 97.8%

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

   2. Simplified 97.8%

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

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

   if -5e-206 < (-.f64 (*.f64 y z) (*.f64 t z)) < 4.99999999999999969e-279

   1. Initial program 76.2%

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

   2. Simplified 99.9%

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

      [Start] 76.2%	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-*l/ [<=] 76.2%	\[ \color{blue}{\frac{x}{y \cdot z - t \cdot z} \cdot 2} \]
      *-commutative [=>] 76.2%	\[ \color{blue}{2 \cdot \frac{x}{y \cdot z - t \cdot z}} \]
      distribute-rgt-out-- [=>] 80.6%	\[ 2 \cdot \frac{x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 99.9%	\[ 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y - t}} \]

   3. Applied egg-rr 100.0%

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

      [Start] 99.9%	\[ 2 \cdot \frac{\frac{x}{z}}{y - t} \]
      associate-*r/ [=>] 99.9%	\[ \color{blue}{\frac{2 \cdot \frac{x}{z}}{y - t}} \]
      associate-*l/ [<=] 99.8%	\[ \color{blue}{\frac{2}{y - t} \cdot \frac{x}{z}} \]
      clear-num [=>] 99.9%	\[ \frac{2}{y - t} \cdot \color{blue}{\frac{1}{\frac{z}{x}}} \]
      un-div-inv [=>] 100.0%	\[ \color{blue}{\frac{\frac{2}{y - t}}{\frac{z}{x}}} \]

   if 4.99999999999999969e-279 < (-.f64 (*.f64 y z) (*.f64 t z)) < 4.99999999999999989e194

   1. Initial program 99.7%

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

   if 4.99999999999999989e194 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 74.1%

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

   2. Simplified 76.4%

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

      [Start] 74.1%	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 76.4%	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\frac{z}{2} \cdot \frac{y - t}{x}\right)}^{-1}} \]
      Step-by-step derivation

      [Start] 76.4%	\[ \frac{x \cdot 2}{z \cdot \left(y - t\right)} \]
      clear-num [=>] 76.4%	\[ \color{blue}{\frac{1}{\frac{z \cdot \left(y - t\right)}{x \cdot 2}}} \]
      inv-pow [=>] 76.4%	\[ \color{blue}{{\left(\frac{z \cdot \left(y - t\right)}{x \cdot 2}\right)}^{-1}} \]
      *-commutative [=>] 76.4%	\[ {\left(\frac{z \cdot \left(y - t\right)}{\color{blue}{2 \cdot x}}\right)}^{-1} \]
      times-frac [=>] 99.9%	\[ {\color{blue}{\left(\frac{z}{2} \cdot \frac{y - t}{x}\right)}}^{-1} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{2} \cdot \frac{y - t}{x}}} \]
      Step-by-step derivation

      [Start] 99.9%	\[ {\left(\frac{z}{2} \cdot \frac{y - t}{x}\right)}^{-1} \]
      unpow-1 [=>] 99.9%	\[ \color{blue}{\frac{1}{\frac{z}{2} \cdot \frac{y - t}{x}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -5 \cdot 10^{-206}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 5 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{2}{y - t}}{\frac{z}{x}}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 5 \cdot 10^{+194}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{2} \cdot \frac{y - t}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.3%
Cost	2252

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-206}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{2}{y - t}}{\frac{z}{x}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+194}:\\ \;\;\;\;\frac{x \cdot 2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{2} \cdot \frac{y - t}{x}}\\ \end{array} \]

Alternative 2
Accuracy	74.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-55} \lor \neg \left(y \leq 2.6 \cdot 10^{-27}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \end{array} \]

Alternative 3
Accuracy	74.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-55} \lor \neg \left(y \leq 1.18 \cdot 10^{-26}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]

Alternative 4
Accuracy	74.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{-55} \lor \neg \left(y \leq 2.4 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]

Alternative 5
Accuracy	74.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-56} \lor \neg \left(y \leq 2.6 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\frac{z}{x}}}{t}\\ \end{array} \]

Alternative 6
Accuracy	93.4%
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-121}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \end{array} \]

Alternative 7
Accuracy	93.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-121}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t - y}}{z}\\ \end{array} \]

Alternative 8
Accuracy	93.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-120}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y - t}{\frac{2}{z}}}\\ \end{array} \]

Alternative 9
Accuracy	93.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-120}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \end{array} \]

Alternative 10
Accuracy	92.0%
Cost	576

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

Alternative 11
Accuracy	53.0%
Cost	448

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

Reproduce

herbie shell --seed 2023263 
(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))))