Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 7.1 → 0.6

Time: 4.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{y - t}\\ t_2 := y \cdot z - z \cdot t\\ t_3 := 2 \cdot \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+230}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{z}{t_1}}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-244}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{t_1}{z}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{y - t} \cdot \frac{x}{z}\right)\\ \end{array} \]

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 (- y t)))
        (t_2 (- (* y z) (* z t)))
        (t_3 (* 2.0 (/ x (* z (- y t))))))
   (if (<= t_2 -1e+230)
     (* 2.0 (/ 1.0 (/ z t_1)))
     (if (<= t_2 -1e-244)
       t_3
       (if (<= t_2 2e-106)
         (* 2.0 (/ t_1 z))
         (if (<= t_2 5e+264) t_3 (* 2.0 (* (/ 1.0 (- y t)) (/ x 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 = x / (y - t);
	double t_2 = (y * z) - (z * t);
	double t_3 = 2.0 * (x / (z * (y - t)));
	double tmp;
	if (t_2 <= -1e+230) {
		tmp = 2.0 * (1.0 / (z / t_1));
	} else if (t_2 <= -1e-244) {
		tmp = t_3;
	} else if (t_2 <= 2e-106) {
		tmp = 2.0 * (t_1 / z);
	} else if (t_2 <= 5e+264) {
		tmp = t_3;
	} else {
		tmp = 2.0 * ((1.0 / (y - t)) * (x / 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = x / (y - t)
    t_2 = (y * z) - (z * t)
    t_3 = 2.0d0 * (x / (z * (y - t)))
    if (t_2 <= (-1d+230)) then
        tmp = 2.0d0 * (1.0d0 / (z / t_1))
    else if (t_2 <= (-1d-244)) then
        tmp = t_3
    else if (t_2 <= 2d-106) then
        tmp = 2.0d0 * (t_1 / z)
    else if (t_2 <= 5d+264) then
        tmp = t_3
    else
        tmp = 2.0d0 * ((1.0d0 / (y - t)) * (x / 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 = x / (y - t);
	double t_2 = (y * z) - (z * t);
	double t_3 = 2.0 * (x / (z * (y - t)));
	double tmp;
	if (t_2 <= -1e+230) {
		tmp = 2.0 * (1.0 / (z / t_1));
	} else if (t_2 <= -1e-244) {
		tmp = t_3;
	} else if (t_2 <= 2e-106) {
		tmp = 2.0 * (t_1 / z);
	} else if (t_2 <= 5e+264) {
		tmp = t_3;
	} else {
		tmp = 2.0 * ((1.0 / (y - t)) * (x / 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 = x / (y - t)
	t_2 = (y * z) - (z * t)
	t_3 = 2.0 * (x / (z * (y - t)))
	tmp = 0
	if t_2 <= -1e+230:
		tmp = 2.0 * (1.0 / (z / t_1))
	elif t_2 <= -1e-244:
		tmp = t_3
	elif t_2 <= 2e-106:
		tmp = 2.0 * (t_1 / z)
	elif t_2 <= 5e+264:
		tmp = t_3
	else:
		tmp = 2.0 * ((1.0 / (y - t)) * (x / 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(x / Float64(y - t))
	t_2 = Float64(Float64(y * z) - Float64(z * t))
	t_3 = Float64(2.0 * Float64(x / Float64(z * Float64(y - t))))
	tmp = 0.0
	if (t_2 <= -1e+230)
		tmp = Float64(2.0 * Float64(1.0 / Float64(z / t_1)));
	elseif (t_2 <= -1e-244)
		tmp = t_3;
	elseif (t_2 <= 2e-106)
		tmp = Float64(2.0 * Float64(t_1 / z));
	elseif (t_2 <= 5e+264)
		tmp = t_3;
	else
		tmp = Float64(2.0 * Float64(Float64(1.0 / Float64(y - t)) * Float64(x / 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 = x / (y - t);
	t_2 = (y * z) - (z * t);
	t_3 = 2.0 * (x / (z * (y - t)));
	tmp = 0.0;
	if (t_2 <= -1e+230)
		tmp = 2.0 * (1.0 / (z / t_1));
	elseif (t_2 <= -1e-244)
		tmp = t_3;
	elseif (t_2 <= 2e-106)
		tmp = 2.0 * (t_1 / z);
	elseif (t_2 <= 5e+264)
		tmp = t_3;
	else
		tmp = 2.0 * ((1.0 / (y - t)) * (x / 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[(x / N[(y - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * z), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+230], N[(2.0 * N[(1.0 / N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-244], t$95$3, If[LessEqual[t$95$2, 2e-106], N[(2.0 * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+264], t$95$3, N[(2.0 * N[(N[(1.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	7.1
Target	2.1
Herbie	0.6

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -1.0000000000000001e230

   1. Initial program 13.2

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.2

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{1}{y - t} \cdot \frac{x}{z}\right)} \]

   4. Applied egg-rr 0.8

      \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{z}{\frac{x}{y - t}}}} \]

   if -1.0000000000000001e230 < (-.f64 (*.f64 y z) (*.f64 t z)) < -9.9999999999999993e-245 or 1.99999999999999988e-106 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.00000000000000033e264

   1. Initial program 0.2

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

   2. Simplified 8.7

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

   3. Applied egg-rr 0.4

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot \left(\frac{1}{z} \cdot \frac{1}{y - t}\right)\right)} \]

   4. Taylor expanded in x around 0 0.2

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

   if -9.9999999999999993e-245 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1.99999999999999988e-106

   1. Initial program 11.8

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

   2. Simplified 3.0

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

   3. Applied egg-rr 3.1

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{1}{y - t} \cdot \frac{x}{z}\right)} \]

   4. Applied egg-rr 3.4

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

   if 5.00000000000000033e264 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 23.6

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{1}{y - t} \cdot \frac{x}{z}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -1 \cdot 10^{+230}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{z}{\frac{x}{y - t}}}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq -1 \cdot 10^{-244}:\\ \;\;\;\;2 \cdot \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 2 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 5 \cdot 10^{+264}:\\ \;\;\;\;2 \cdot \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{y - t} \cdot \frac{x}{z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))))