Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.9 → 1.8

Time: 19.3s

Precision: binary64

Cost: 2640

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - t \cdot z\\ t_2 := \frac{2 \cdot \frac{x}{z}}{y - t}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+299}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;t_1 \leq 10^{-78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (- (* y z) (* t z))) (t_2 (/ (* 2.0 (/ x z)) (- y t))))
   (if (<= t_1 -2e+299)
     (* (/ x (- y t)) (/ 2.0 z))
     (if (<= t_1 -2e-264)
       (* (/ x (* (- y t) z)) 2.0)
       (if (<= t_1 1e-78)
         t_2
         (if (<= t_1 4e+46) (* x (/ (/ 2.0 (- y t)) z)) t_2))))))

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) - (t * z);
	double t_2 = (2.0 * (x / z)) / (y - t);
	double tmp;
	if (t_1 <= -2e+299) {
		tmp = (x / (y - t)) * (2.0 / z);
	} else if (t_1 <= -2e-264) {
		tmp = (x / ((y - t) * z)) * 2.0;
	} else if (t_1 <= 1e-78) {
		tmp = t_2;
	} else if (t_1 <= 4e+46) {
		tmp = x * ((2.0 / (y - t)) / z);
	} else {
		tmp = t_2;
	}
	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 = (y * z) - (t * z)
    t_2 = (2.0d0 * (x / z)) / (y - t)
    if (t_1 <= (-2d+299)) then
        tmp = (x / (y - t)) * (2.0d0 / z)
    else if (t_1 <= (-2d-264)) then
        tmp = (x / ((y - t) * z)) * 2.0d0
    else if (t_1 <= 1d-78) then
        tmp = t_2
    else if (t_1 <= 4d+46) then
        tmp = x * ((2.0d0 / (y - t)) / z)
    else
        tmp = t_2
    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) - (t * z);
	double t_2 = (2.0 * (x / z)) / (y - t);
	double tmp;
	if (t_1 <= -2e+299) {
		tmp = (x / (y - t)) * (2.0 / z);
	} else if (t_1 <= -2e-264) {
		tmp = (x / ((y - t) * z)) * 2.0;
	} else if (t_1 <= 1e-78) {
		tmp = t_2;
	} else if (t_1 <= 4e+46) {
		tmp = x * ((2.0 / (y - t)) / z);
	} else {
		tmp = t_2;
	}
	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) - (t * z)
	t_2 = (2.0 * (x / z)) / (y - t)
	tmp = 0
	if t_1 <= -2e+299:
		tmp = (x / (y - t)) * (2.0 / z)
	elif t_1 <= -2e-264:
		tmp = (x / ((y - t) * z)) * 2.0
	elif t_1 <= 1e-78:
		tmp = t_2
	elif t_1 <= 4e+46:
		tmp = x * ((2.0 / (y - t)) / z)
	else:
		tmp = t_2
	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(t * z))
	t_2 = Float64(Float64(2.0 * Float64(x / z)) / Float64(y - t))
	tmp = 0.0
	if (t_1 <= -2e+299)
		tmp = Float64(Float64(x / Float64(y - t)) * Float64(2.0 / z));
	elseif (t_1 <= -2e-264)
		tmp = Float64(Float64(x / Float64(Float64(y - t) * z)) * 2.0);
	elseif (t_1 <= 1e-78)
		tmp = t_2;
	elseif (t_1 <= 4e+46)
		tmp = Float64(x * Float64(Float64(2.0 / Float64(y - t)) / z));
	else
		tmp = t_2;
	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) - (t * z);
	t_2 = (2.0 * (x / z)) / (y - t);
	tmp = 0.0;
	if (t_1 <= -2e+299)
		tmp = (x / (y - t)) * (2.0 / z);
	elseif (t_1 <= -2e-264)
		tmp = (x / ((y - t) * z)) * 2.0;
	elseif (t_1 <= 1e-78)
		tmp = t_2;
	elseif (t_1 <= 4e+46)
		tmp = x * ((2.0 / (y - t)) / z);
	else
		tmp = t_2;
	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[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+299], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-264], N[(N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e-78], t$95$2, If[LessEqual[t$95$1, 4e+46], N[(x * N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Target

Original	6.9
Target	2.3
Herbie	1.8

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -2.0000000000000001e299

   1. Initial program 18.5

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

   2. Simplified 18.5

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

      [Start] 18.5	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      rational.json-simplify-2 [=>] 18.5	\[ \frac{x \cdot 2}{\color{blue}{z \cdot y} - t \cdot z} \]
      rational.json-simplify-52 [=>] 18.5	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]

   3. Applied egg-rr 0.1

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

   if -2.0000000000000001e299 < (-.f64 (*.f64 y z) (*.f64 t z)) < -2e-264

   1. Initial program 0.2

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

   2. Simplified 8.2

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

      [Start] 0.2	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      rational.json-simplify-2 [=>] 0.2	\[ \frac{x \cdot 2}{\color{blue}{z \cdot y} - t \cdot z} \]
      rational.json-simplify-52 [=>] 0.2	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      rational.json-simplify-46 [=>] 8.2	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]
      rational.json-simplify-49 [=>] 8.2	\[ \frac{\color{blue}{2 \cdot \frac{x}{z}}}{y - t} \]

   3. Applied egg-rr 7.9

      \[\leadsto \color{blue}{\frac{\frac{2}{y - t}}{\frac{z}{x}} + 0} \]

   4. Simplified 8.2

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

      [Start] 7.9	\[ \frac{\frac{2}{y - t}}{\frac{z}{x}} + 0 \]
      rational.json-simplify-4 [=>] 7.9	\[ \color{blue}{\frac{\frac{2}{y - t}}{\frac{z}{x}}} \]
      rational.json-simplify-47 [=>] 8.2	\[ \color{blue}{\frac{2}{\left(y - t\right) \cdot \frac{z}{x}}} \]

   5. Applied egg-rr 0.2

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

   if -2e-264 < (-.f64 (*.f64 y z) (*.f64 t z)) < 9.99999999999999999e-79 or 4e46 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 10.9

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

   2. Simplified 3.9

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

      [Start] 10.9	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      rational.json-simplify-2 [=>] 10.9	\[ \frac{x \cdot 2}{\color{blue}{z \cdot y} - t \cdot z} \]
      rational.json-simplify-52 [=>] 8.5	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      rational.json-simplify-46 [=>] 3.9	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]
      rational.json-simplify-49 [=>] 3.9	\[ \frac{\color{blue}{2 \cdot \frac{x}{z}}}{y - t} \]

   if 9.99999999999999999e-79 < (-.f64 (*.f64 y z) (*.f64 t z)) < 4e46

   1. Initial program 0.4

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

   2. Simplified 0.4

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

      [Start] 0.4	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      rational.json-simplify-2 [=>] 0.4	\[ \frac{\color{blue}{2 \cdot x}}{y \cdot z - t \cdot z} \]
      rational.json-simplify-49 [=>] 0.3	\[ \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]
      rational.json-simplify-2 [=>] 0.3	\[ x \cdot \frac{2}{\color{blue}{z \cdot y} - t \cdot z} \]
      rational.json-simplify-52 [=>] 0.3	\[ x \cdot \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      rational.json-simplify-46 [=>] 0.3	\[ x \cdot \color{blue}{\frac{\frac{2}{z}}{y - t}} \]
      rational.json-simplify-44 [=>] 0.4	\[ x \cdot \color{blue}{\frac{\frac{2}{y - t}}{z}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - t \cdot z \leq -2 \cdot 10^{+299}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \mathbf{elif}\;y \cdot z - t \cdot z \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;y \cdot z - t \cdot z \leq 10^{-78}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - t \cdot z \leq 4 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{z}}{y - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.4
Cost	1096

\[\begin{array}{l} t_1 := \frac{x}{y - t} \cdot \frac{2}{z}\\ \mathbf{if}\;x \cdot 2 \leq -4 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot 2 \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	2.7
Cost	840

\[\begin{array}{l} t_1 := \frac{2}{\left(y - t\right) \cdot \frac{z}{x}}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	2.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\left(y - t\right) \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{y - t}{\frac{x}{z}}}\\ \end{array} \]

Alternative 4
Error	17.6
Cost	712

\[\begin{array}{l} t_1 := x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{if}\;y \leq -2.55 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-65}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	17.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-65}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]

Alternative 6
Error	17.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{-2}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]

Alternative 7
Error	17.4
Cost	712

\[\begin{array}{l} t_1 := \frac{\frac{2}{z}}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{-2}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	17.2
Cost	712

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{z \cdot 0.5}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{-2}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	6.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+184}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{\frac{z}{-2}}\\ \end{array} \]

Alternative 10
Error	31.5
Cost	448

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

Alternative 11
Error	31.4
Cost	448

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

Reproduce

herbie shell --seed 2023074 
(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))))