Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.3 → 1.4

Time: 4.0s

Precision: binary64

\[ \begin{array}{c}[y, t] = \mathsf{sort}([y, t])\\ \end{array} \]

Math

\[\left(x \cdot y - z \cdot y\right) \cdot t \]

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(y, -z, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right) - y \cdot \left(z \cdot t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (- (* x y) (* z y)) t))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x y) (* y z))))
   (if (<= t_1 (- INFINITY))
     (* y (* t (- x z)))
     (if (<= t_1 2e+232)
       (fma (* y (- x z)) t (* t (fma y (- z) (* y z))))
       (- (* y (* x t)) (* y (* z t)))))))

C

double code(double x, double y, double z, double t) {
	return ((x * y) - (z * y)) * t;
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = (x * y) - (y * z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (t * (x - z));
	} else if (t_1 <= 2e+232) {
		tmp = fma((y * (x - z)), t, (t * fma(y, -z, (y * z))));
	} else {
		tmp = (y * (x * t)) - (y * (z * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * y) - Float64(z * y)) * t)
end

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(x * y) - Float64(y * z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(t * Float64(x - z)));
	elseif (t_1 <= 2e+232)
		tmp = fma(Float64(y * Float64(x - z)), t, Float64(t * fma(y, Float64(-z), Float64(y * z))));
	else
		tmp = Float64(Float64(y * Float64(x * t)) - Float64(y * Float64(z * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * y), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+232], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t + N[(t * N[(y * (-z) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision] - N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.3
Target	3.4
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -inf.0

   1. Initial program 64.0

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

   2. Simplified 0.3

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

   3. Applied egg-rr 0.3

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z y)) < 2.00000000000000011e232

   1. Initial program 1.7

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

   2. Applied egg-rr 1.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(x - z\right), t, \mathsf{fma}\left(y, -z, y \cdot z\right) \cdot t\right)} \]

   if 2.00000000000000011e232 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 37.2

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

   2. Simplified 0.5

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

   3. Taylor expanded in x around 0 0.5

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -\infty:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 2 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(y, -z, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right) - y \cdot \left(z \cdot t\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

  (* (- (* x y) (* z y)) t))