Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Accuracy: 89.7% → 96.0%

Time: 15.5s

Precision: binary64

Cost: 1097

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot 2 \leq -5 \cdot 10^{-97} \lor \neg \left(x \cdot 2 \leq 2 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x 2.0) -5e-97) (not (<= (* x 2.0) 2e-43)))
   (/ (/ x (- y t)) (* z 0.5))
   (/ (/ (* x 2.0) z) (- y t))))

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 tmp;
	if (((x * 2.0) <= -5e-97) || !((x * 2.0) <= 2e-43)) {
		tmp = (x / (y - t)) / (z * 0.5);
	} else {
		tmp = ((x * 2.0) / z) / (y - t);
	}
	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) :: tmp
    if (((x * 2.0d0) <= (-5d-97)) .or. (.not. ((x * 2.0d0) <= 2d-43))) then
        tmp = (x / (y - t)) / (z * 0.5d0)
    else
        tmp = ((x * 2.0d0) / z) / (y - t)
    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 tmp;
	if (((x * 2.0) <= -5e-97) || !((x * 2.0) <= 2e-43)) {
		tmp = (x / (y - t)) / (z * 0.5);
	} else {
		tmp = ((x * 2.0) / z) / (y - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (x * 2.0) / ((y * z) - (t * z))

↓

def code(x, y, z, t):
	tmp = 0
	if ((x * 2.0) <= -5e-97) or not ((x * 2.0) <= 2e-43):
		tmp = (x / (y - t)) / (z * 0.5)
	else:
		tmp = ((x * 2.0) / z) / (y - t)
	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)
	tmp = 0.0
	if ((Float64(x * 2.0) <= -5e-97) || !(Float64(x * 2.0) <= 2e-43))
		tmp = Float64(Float64(x / Float64(y - t)) / Float64(z * 0.5));
	else
		tmp = Float64(Float64(Float64(x * 2.0) / z) / Float64(y - t));
	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)
	tmp = 0.0;
	if (((x * 2.0) <= -5e-97) || ~(((x * 2.0) <= 2e-43)))
		tmp = (x / (y - t)) / (z * 0.5);
	else
		tmp = ((x * 2.0) / z) / (y - t);
	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_] := If[Or[LessEqual[N[(x * 2.0), $MachinePrecision], -5e-97], N[Not[LessEqual[N[(x * 2.0), $MachinePrecision], 2e-43]], $MachinePrecision]], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]]

Target

Original	89.7%
Target	96.6%
Herbie	96.0%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 2) < -4.9999999999999995e-97 or 2.00000000000000015e-43 < (*.f64 x 2)

   1. Initial program 86.7%

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

   2. Simplified 96.0%

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

      [Start] 86.7	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      *-commutative [=>] 86.7	\[ \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 88.3	\[ \frac{2 \cdot x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      times-frac [=>] 95.8	\[ \color{blue}{\frac{2}{z} \cdot \frac{x}{y - t}} \]
      associate-*r/ [=>] 86.3	\[ \color{blue}{\frac{\frac{2}{z} \cdot x}{y - t}} \]
      associate-/l* [=>] 96.0	\[ \color{blue}{\frac{\frac{2}{z}}{\frac{y - t}{x}}} \]

   3. Applied egg-rr 95.5%

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

      [Start] 96.0	\[ \frac{\frac{2}{z}}{\frac{y - t}{x}} \]
      clear-num [=>] 95.4	\[ \color{blue}{\frac{1}{\frac{\frac{y - t}{x}}{\frac{2}{z}}}} \]
      inv-pow [=>] 95.4	\[ \color{blue}{{\left(\frac{\frac{y - t}{x}}{\frac{2}{z}}\right)}^{-1}} \]
      div-inv [=>] 95.4	\[ {\color{blue}{\left(\frac{y - t}{x} \cdot \frac{1}{\frac{2}{z}}\right)}}^{-1} \]
      clear-num [<=] 95.5	\[ {\left(\frac{y - t}{x} \cdot \color{blue}{\frac{z}{2}}\right)}^{-1} \]
      div-inv [=>] 95.5	\[ {\left(\frac{y - t}{x} \cdot \color{blue}{\left(z \cdot \frac{1}{2}\right)}\right)}^{-1} \]
      metadata-eval [=>] 95.5	\[ {\left(\frac{y - t}{x} \cdot \left(z \cdot \color{blue}{0.5}\right)\right)}^{-1} \]

   4. Simplified 95.9%

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

      [Start] 95.5	\[ {\left(\frac{y - t}{x} \cdot \left(z \cdot 0.5\right)\right)}^{-1} \]
      unpow-1 [=>] 95.5	\[ \color{blue}{\frac{1}{\frac{y - t}{x} \cdot \left(z \cdot 0.5\right)}} \]
      associate-/r* [=>] 95.6	\[ \color{blue}{\frac{\frac{1}{\frac{y - t}{x}}}{z \cdot 0.5}} \]
      *-lft-identity [<=] 95.6	\[ \frac{\frac{1}{\color{blue}{1 \cdot \frac{y - t}{x}}}}{z \cdot 0.5} \]
      associate-*r/ [=>] 95.6	\[ \frac{\frac{1}{\color{blue}{\frac{1 \cdot \left(y - t\right)}{x}}}}{z \cdot 0.5} \]
      associate-*l/ [<=] 95.5	\[ \frac{\frac{1}{\color{blue}{\frac{1}{x} \cdot \left(y - t\right)}}}{z \cdot 0.5} \]
      associate-/r* [=>] 95.8	\[ \frac{\color{blue}{\frac{\frac{1}{\frac{1}{x}}}{y - t}}}{z \cdot 0.5} \]
      associate-/r/ [=>] 95.9	\[ \frac{\frac{\color{blue}{\frac{1}{1} \cdot x}}{y - t}}{z \cdot 0.5} \]
      metadata-eval [=>] 95.9	\[ \frac{\frac{\color{blue}{1} \cdot x}{y - t}}{z \cdot 0.5} \]
      associate-*r/ [<=] 95.9	\[ \frac{\color{blue}{1 \cdot \frac{x}{y - t}}}{z \cdot 0.5} \]
      *-lft-identity [=>] 95.9	\[ \frac{\color{blue}{\frac{x}{y - t}}}{z \cdot 0.5} \]

   if -4.9999999999999995e-97 < (*.f64 x 2) < 2.00000000000000015e-43

   1. Initial program 93.9%

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

   2. Simplified 96.1%

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

      [Start] 93.9	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 96.4	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 96.1	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq -5 \cdot 10^{-97} \lor \neg \left(x \cdot 2 \leq 2 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+82} \lor \neg \left(z \leq 7 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\left(y - t\right) \cdot z}{2}}\\ \end{array} \]

Alternative 2
Accuracy	96.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-44} \lor \neg \left(x \leq 2.2 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \end{array} \]

Alternative 3
Accuracy	73.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-48} \lor \neg \left(y \leq 1.78 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{t \cdot z}\\ \end{array} \]

Alternative 4
Accuracy	73.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{-48} \lor \neg \left(y \leq 2.95 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \end{array} \]

Alternative 5
Accuracy	73.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 6
Accuracy	73.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+23}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 7
Accuracy	73.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 8
Accuracy	73.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.62 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 9
Accuracy	91.9%
Cost	576

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

Alternative 10
Accuracy	50.5%
Cost	448

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

Reproduce

herbie shell --seed 2023137 
(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))))