Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 7.0 → 1.2

Time: 11.5s

Precision: binary64

Cost: 1608

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+219}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{elif}\;t_1 \leq 10^{+302}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \end{array} \]

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 -1e+219)
     (/ (/ (* x 2.0) z) (- y t))
     (if (<= t_1 1e+302)
       (/ x (/ (* z (- y t)) 2.0))
       (/ (/ x (- y t)) (* z 0.5))))))

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 <= -1e+219) {
		tmp = ((x * 2.0) / z) / (y - t);
	} else if (t_1 <= 1e+302) {
		tmp = x / ((z * (y - t)) / 2.0);
	} else {
		tmp = (x / (y - t)) / (z * 0.5);
	}
	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 <= (-1d+219)) then
        tmp = ((x * 2.0d0) / z) / (y - t)
    else if (t_1 <= 1d+302) then
        tmp = x / ((z * (y - t)) / 2.0d0)
    else
        tmp = (x / (y - t)) / (z * 0.5d0)
    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 <= -1e+219) {
		tmp = ((x * 2.0) / z) / (y - t);
	} else if (t_1 <= 1e+302) {
		tmp = x / ((z * (y - t)) / 2.0);
	} else {
		tmp = (x / (y - t)) / (z * 0.5);
	}
	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 <= -1e+219:
		tmp = ((x * 2.0) / z) / (y - t)
	elif t_1 <= 1e+302:
		tmp = x / ((z * (y - t)) / 2.0)
	else:
		tmp = (x / (y - t)) / (z * 0.5)
	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 <= -1e+219)
		tmp = Float64(Float64(Float64(x * 2.0) / z) / Float64(y - t));
	elseif (t_1 <= 1e+302)
		tmp = Float64(x / Float64(Float64(z * Float64(y - t)) / 2.0));
	else
		tmp = Float64(Float64(x / Float64(y - t)) / Float64(z * 0.5));
	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 <= -1e+219)
		tmp = ((x * 2.0) / z) / (y - t);
	elseif (t_1 <= 1e+302)
		tmp = x / ((z * (y - t)) / 2.0);
	else
		tmp = (x / (y - t)) / (z * 0.5);
	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, -1e+219], N[(N[(N[(x * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+302], N[(x / N[(N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.0
Target	2.1
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -9.99999999999999965e218

   1. Initial program 12.6

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

   2. Simplified 0.5

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

      [Start] 12.6	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      distribute-rgt-out-- [=>] 12.6	\[ \frac{x \cdot 2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 0.5	\[ \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}} \]

   if -9.99999999999999965e218 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1.0000000000000001e302

   1. Initial program 1.6

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

   2. Simplified 1.5

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

      [Start] 1.6	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-/l* [=>] 1.5	\[ \color{blue}{\frac{x}{\frac{y \cdot z - t \cdot z}{2}}} \]
      distribute-rgt-out-- [=>] 1.5	\[ \frac{x}{\frac{\color{blue}{z \cdot \left(y - t\right)}}{2}} \]

   if 1.0000000000000001e302 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 28.4

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

   2. Simplified 18.5

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

      [Start] 28.4	\[ \frac{x \cdot 2}{y \cdot z - t \cdot z} \]
      associate-*r/ [<=] 28.4	\[ \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]
      distribute-rgt-out-- [=>] 19.7	\[ x \cdot \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \]
      associate-/r* [=>] 18.5	\[ x \cdot \color{blue}{\frac{\frac{2}{z}}{y - t}} \]

   3. Applied egg-rr 19.7

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

   4. Simplified 0.1

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

      [Start] 19.7	\[ \frac{x}{\left(y - t\right) \cdot \left(z \cdot 0.5\right)} \]
      associate-/r* [=>] 0.1	\[ \color{blue}{\frac{\frac{x}{y - t}}{z \cdot 0.5}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -1 \cdot 10^{+219}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 10^{+302}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.0
Cost	1241

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{-2}}}{t}\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{-105} \lor \neg \left(y \leq 8 \cdot 10^{-35}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \end{array} \]

Alternative 2
Error	18.1
Cost	1240

\[\begin{array}{l} t_1 := \frac{\frac{x \cdot 2}{z}}{y}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{-2}}}{t}\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{-105}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{-34}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	2.2
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \cdot 2 \leq -2 \cdot 10^{+59} \lor \neg \left(x \cdot 2 \leq 10^{-31}\right):\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 4
Error	2.3
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \cdot 2 \leq -2 \cdot 10^{+59} \lor \neg \left(x \cdot 2 \leq 5 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \end{array} \]

Alternative 5
Error	18.7
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+85} \lor \neg \left(y \leq -7.8 \cdot 10^{+61}\right) \land \left(y \leq -6.6 \cdot 10^{-105} \lor \neg \left(y \leq 1.46 \cdot 10^{-35}\right)\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \end{array} \]

Alternative 6
Error	18.6
Cost	977

\[\begin{array}{l} t_1 := x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;\neg \left(y \leq -6.6 \cdot 10^{-105}\right) \land y \leq 7.4 \cdot 10^{-35}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	18.6
Cost	976

\[\begin{array}{l} t_1 := x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 4.05 \cdot 10^{-35}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	18.5
Cost	976

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-35}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 9
Error	6.0
Cost	576

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

Alternative 10
Error	5.7
Cost	576

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

Alternative 11
Error	31.3
Cost	448

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

Reproduce

herbie shell --seed 2023031 
(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))))