Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.8 → 2.1

Time: 13.8s

Precision: binary64

Cost: 2120

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot 2}{y \cdot z - z \cdot t}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-297}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

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+71)
     (/ (/ 2.0 z) (/ (- y t) x))
     (if (<= t_1 2e-297)
       (/ (/ (* x 2.0) z) (- y t))
       (/ x (/ (* z (- y t)) 2.0))))))

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+71) {
		tmp = (2.0 / z) / ((y - t) / x);
	} else if (t_1 <= 2e-297) {
		tmp = ((x * 2.0) / z) / (y - t);
	} else {
		tmp = x / ((z * (y - t)) / 2.0);
	}
	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+71)) then
        tmp = (2.0d0 / z) / ((y - t) / x)
    else if (t_1 <= 2d-297) then
        tmp = ((x * 2.0d0) / z) / (y - t)
    else
        tmp = x / ((z * (y - t)) / 2.0d0)
    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+71) {
		tmp = (2.0 / z) / ((y - t) / x);
	} else if (t_1 <= 2e-297) {
		tmp = ((x * 2.0) / z) / (y - t);
	} else {
		tmp = x / ((z * (y - t)) / 2.0);
	}
	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+71:
		tmp = (2.0 / z) / ((y - t) / x)
	elif t_1 <= 2e-297:
		tmp = ((x * 2.0) / z) / (y - t)
	else:
		tmp = x / ((z * (y - t)) / 2.0)
	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+71)
		tmp = Float64(Float64(2.0 / z) / Float64(Float64(y - t) / x));
	elseif (t_1 <= 2e-297)
		tmp = Float64(Float64(Float64(x * 2.0) / z) / Float64(y - t));
	else
		tmp = Float64(x / Float64(Float64(z * Float64(y - t)) / 2.0));
	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+71)
		tmp = (2.0 / z) / ((y - t) / x);
	elseif (t_1 <= 2e-297)
		tmp = ((x * 2.0) / z) / (y - t);
	else
		tmp = x / ((z * (y - t)) / 2.0);
	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+71], N[(N[(2.0 / z), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-297], N[(N[(N[(x * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.8
Target	2.0
Herbie	2.1

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z))) < -2.0000000000000001e71

   1. Initial program 4.7

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

   2. Simplified 4.3

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

      [Start] 4.7	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      *-commutative [=>] 4.7	\[ \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 4.7	\[ \frac{2 \cdot x}{\color{blue}{z \cdot \left(y - t\right)}} \]
      times-frac [=>] 4.8	\[ \color{blue}{\frac{2}{z} \cdot \frac{x}{y - t}} \]
      associate-*r/ [=>] 19.9	\[ \color{blue}{\frac{\frac{2}{z} \cdot x}{y - t}} \]
      associate-/l* [=>] 4.3	\[ \color{blue}{\frac{\frac{2}{z}}{\frac{y - t}{x}}} \]

   if -2.0000000000000001e71 < (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z))) < 2.00000000000000008e-297

   1. Initial program 7.9

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

   2. Simplified 1.3

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

      [Start] 7.9	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 7.9	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 1.3	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]

   if 2.00000000000000008e-297 < (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 5.8

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

   2. Simplified 2.6

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

      [Start] 5.8	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-/l* [=>] 5.8	\[ \color{blue}{\frac{x}{\frac{y \cdot z - t \cdot z}{2}}} \]
      distribute-rgt-out-- [=>] 2.6	\[ \frac{x}{\frac{\color{blue}{z \cdot \left(y - t\right)}}{2}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -2 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq 2 \cdot 10^{-297}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.0
Cost	1357

\[\begin{array}{l} \mathbf{if}\;x \cdot 2 \leq -6 \cdot 10^{-10} \lor \neg \left(x \cdot 2 \leq 5 \cdot 10^{-50}\right) \land x \cdot 2 \leq 2 \cdot 10^{+234}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 2
Error	17.5
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-191}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-36}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 3
Error	17.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-192}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 4
Error	17.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-192}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 5
Error	17.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-193}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 6
Error	17.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-198}:\\ \;\;\;\;\frac{-2}{\frac{z \cdot t}{x}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-38}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 7
Error	17.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{-194}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 8
Error	17.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-192}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-36}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]

Alternative 9
Error	17.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-193}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{x \cdot \frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]

Alternative 10
Error	5.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-236} \lor \neg \left(t \leq 2.4 \cdot 10^{-230}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 11
Error	2.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -205 \lor \neg \left(z \leq 5 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 12
Error	17.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+29} \lor \neg \left(y \leq 2.6 \cdot 10^{-37}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]

Alternative 13
Error	17.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]

Alternative 14
Error	17.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{x \cdot \frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]

Alternative 15
Error	4.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]

Alternative 16
Error	31.0
Cost	448

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

Reproduce

herbie shell --seed 2022364 
(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))))