Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.8 → 1.1

Time: 25.2s

Precision: binary64

Cost: 1736

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - t \cdot z\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{elif}\;t_1 \leq 10^{+222}:\\ \;\;\;\;\frac{x \cdot 2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]

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))))
   (if (<= t_1 -2e+306)
     (* (/ x z) (/ 2.0 (- y t)))
     (if (<= t_1 1e+222) (/ (* x 2.0) t_1) (* (/ x (- y t)) (/ 2.0 z))))))

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 tmp;
	if (t_1 <= -2e+306) {
		tmp = (x / z) * (2.0 / (y - t));
	} else if (t_1 <= 1e+222) {
		tmp = (x * 2.0) / t_1;
	} else {
		tmp = (x / (y - t)) * (2.0 / z);
	}
	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) - (t * z)
    if (t_1 <= (-2d+306)) then
        tmp = (x / z) * (2.0d0 / (y - t))
    else if (t_1 <= 1d+222) then
        tmp = (x * 2.0d0) / t_1
    else
        tmp = (x / (y - t)) * (2.0d0 / z)
    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 tmp;
	if (t_1 <= -2e+306) {
		tmp = (x / z) * (2.0 / (y - t));
	} else if (t_1 <= 1e+222) {
		tmp = (x * 2.0) / t_1;
	} else {
		tmp = (x / (y - t)) * (2.0 / z);
	}
	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)
	tmp = 0
	if t_1 <= -2e+306:
		tmp = (x / z) * (2.0 / (y - t))
	elif t_1 <= 1e+222:
		tmp = (x * 2.0) / t_1
	else:
		tmp = (x / (y - t)) * (2.0 / z)
	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))
	tmp = 0.0
	if (t_1 <= -2e+306)
		tmp = Float64(Float64(x / z) * Float64(2.0 / Float64(y - t)));
	elseif (t_1 <= 1e+222)
		tmp = Float64(Float64(x * 2.0) / t_1);
	else
		tmp = Float64(Float64(x / Float64(y - t)) * Float64(2.0 / z));
	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);
	tmp = 0.0;
	if (t_1 <= -2e+306)
		tmp = (x / z) * (2.0 / (y - t));
	elseif (t_1 <= 1e+222)
		tmp = (x * 2.0) / t_1;
	else
		tmp = (x / (y - t)) * (2.0 / z);
	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]}, If[LessEqual[t$95$1, -2e+306], N[(N[(x / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+222], N[(N[(x * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.8
Target	2.0
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -2.00000000000000003e306

   1. Initial program 20.6

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

   2. Simplified 0.1

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

   3. Applied egg-rr 0.1

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

   if -2.00000000000000003e306 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1e222

   1. Initial program 1.4

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

   if 1e222 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 19.3

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

   2. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\frac{x}{y - t} \cdot \frac{2}{z}} \]
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	18.0
Cost	1240

\[\begin{array}{l} t_1 := \frac{-2}{t \cdot z} \cdot x\\ t_2 := \frac{x}{y} \cdot \frac{2}{z}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{y \cdot z} \cdot x\\ \mathbf{elif}\;y \leq -58:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	17.2
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{+26}:\\ \;\;\;\;\frac{2}{y \cdot z} \cdot x\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{2}{z}\\ \end{array} \]

Alternative 3
Error	17.2
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{if}\;y \leq -7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{2}{y \cdot z} \cdot x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{2}{z}\\ \end{array} \]

Alternative 4
Error	6.5
Cost	840

\[\begin{array}{l} t_1 := \frac{2}{\left(y - t\right) \cdot z} \cdot x\\ \mathbf{if}\;z \leq 1.3 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+252}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	3.3
Cost	840

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;\frac{2}{\left(y - t\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	2.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{2}{\left(y - t\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]

Alternative 7
Error	2.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -0.001:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{-2}{t - y}}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]

Alternative 8
Error	2.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -0.000205:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \mathbf{elif}\;z \leq 10^{+62}:\\ \;\;\;\;\frac{\frac{-2}{t - y}}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\left(y - t\right) \cdot 0.5}\\ \end{array} \]

Alternative 9
Error	17.7
Cost	712

\[\begin{array}{l} t_1 := \frac{-2}{t \cdot z} \cdot x\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 680000000000:\\ \;\;\;\;2 \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	17.0
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{if}\;t \leq -1.06 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 680000000000:\\ \;\;\;\;\frac{x}{y} \cdot \frac{2}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	32.0
Cost	448

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

Reproduce

herbie shell --seed 2023010 
(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))))