Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.7 → 0.5

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{x}{z} \cdot 2}{y - t}\\ t_2 := y \cdot z - z \cdot t\\ t_3 := \frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-213}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+203}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 z) 2.0) (- y t)))
        (t_2 (- (* y z) (* z t)))
        (t_3 (/ (* x 2.0) (* z (- y t)))))
   (if (<= t_2 -2e+172)
     t_1
     (if (<= t_2 -1e-213)
       t_3
       (if (<= t_2 5e-291) t_1 (if (<= t_2 1e+203) t_3 t_1))))))

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 / z) * 2.0) / (y - t);
	double t_2 = (y * z) - (z * t);
	double t_3 = (x * 2.0) / (z * (y - t));
	double tmp;
	if (t_2 <= -2e+172) {
		tmp = t_1;
	} else if (t_2 <= -1e-213) {
		tmp = t_3;
	} else if (t_2 <= 5e-291) {
		tmp = t_1;
	} else if (t_2 <= 1e+203) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	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 / z) * 2.0d0) / (y - t)
    t_2 = (y * z) - (z * t)
    t_3 = (x * 2.0d0) / (z * (y - t))
    if (t_2 <= (-2d+172)) then
        tmp = t_1
    else if (t_2 <= (-1d-213)) then
        tmp = t_3
    else if (t_2 <= 5d-291) then
        tmp = t_1
    else if (t_2 <= 1d+203) then
        tmp = t_3
    else
        tmp = t_1
    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 / z) * 2.0) / (y - t);
	double t_2 = (y * z) - (z * t);
	double t_3 = (x * 2.0) / (z * (y - t));
	double tmp;
	if (t_2 <= -2e+172) {
		tmp = t_1;
	} else if (t_2 <= -1e-213) {
		tmp = t_3;
	} else if (t_2 <= 5e-291) {
		tmp = t_1;
	} else if (t_2 <= 1e+203) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	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 / z) * 2.0) / (y - t)
	t_2 = (y * z) - (z * t)
	t_3 = (x * 2.0) / (z * (y - t))
	tmp = 0
	if t_2 <= -2e+172:
		tmp = t_1
	elif t_2 <= -1e-213:
		tmp = t_3
	elif t_2 <= 5e-291:
		tmp = t_1
	elif t_2 <= 1e+203:
		tmp = t_3
	else:
		tmp = t_1
	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(Float64(x / z) * 2.0) / Float64(y - t))
	t_2 = Float64(Float64(y * z) - Float64(z * t))
	t_3 = Float64(Float64(x * 2.0) / Float64(z * Float64(y - t)))
	tmp = 0.0
	if (t_2 <= -2e+172)
		tmp = t_1;
	elseif (t_2 <= -1e-213)
		tmp = t_3;
	elseif (t_2 <= 5e-291)
		tmp = t_1;
	elseif (t_2 <= 1e+203)
		tmp = t_3;
	else
		tmp = t_1;
	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 / z) * 2.0) / (y - t);
	t_2 = (y * z) - (z * t);
	t_3 = (x * 2.0) / (z * (y - t));
	tmp = 0.0;
	if (t_2 <= -2e+172)
		tmp = t_1;
	elseif (t_2 <= -1e-213)
		tmp = t_3;
	elseif (t_2 <= 5e-291)
		tmp = t_1;
	elseif (t_2 <= 1e+203)
		tmp = t_3;
	else
		tmp = t_1;
	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[(N[(x / z), $MachinePrecision] * 2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * z), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * 2.0), $MachinePrecision] / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+172], t$95$1, If[LessEqual[t$95$2, -1e-213], t$95$3, If[LessEqual[t$95$2, 5e-291], t$95$1, If[LessEqual[t$95$2, 1e+203], t$95$3, t$95$1]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.7
Target	2.2
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -2.0000000000000002e172 or -9.9999999999999995e-214 < (-.f64 (*.f64 y z) (*.f64 t z)) < 5.0000000000000003e-291 or 9.9999999999999999e202 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 15.2

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

   2. Simplified 0.7

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

   3. Applied egg-rr 0.8

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

   if -2.0000000000000002e172 < (-.f64 (*.f64 y z) (*.f64 t z)) < -9.9999999999999995e-214 or 5.0000000000000003e-291 < (-.f64 (*.f64 y z) (*.f64 t z)) < 9.9999999999999999e202

   1. Initial program 0.3

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

   2. Simplified 10.1

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

   3. Applied egg-rr 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -2 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 10^{+203}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))