Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 14.7 → 0.9

Time: 1.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ t_1 := \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (* x 2.0) y) (- x y))) (t_1 (/ x (fma 0.5 (/ x y) -0.5))))
   (if (<= t_0 -2e-8)
     t_1
     (if (<= t_0 -5e-307)
       t_0
       (if (<= t_0 0.0) t_1 (if (<= t_0 5e-105) t_0 t_1))))))

C

double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

↓

double code(double x, double y) {
	double t_0 = ((x * 2.0) * y) / (x - y);
	double t_1 = x / fma(0.5, (x / y), -0.5);
	double tmp;
	if (t_0 <= -2e-8) {
		tmp = t_1;
	} else if (t_0 <= -5e-307) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-105) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
end

↓

function code(x, y)
	t_0 = Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
	t_1 = Float64(x / fma(0.5, Float64(x / y), -0.5))
	tmp = 0.0
	if (t_0 <= -2e-8)
		tmp = t_1;
	elseif (t_0 <= -5e-307)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e-105)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(0.5 * N[(x / y), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-8], t$95$1, If[LessEqual[t$95$0, -5e-307], t$95$0, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e-105], t$95$0, t$95$1]]]]]]

Error

Bits error versus x

Bits error versus y

Target

Original	14.7
Target	0.3
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < -2e-8 or -5.00000000000000014e-307 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < 0.0 or 4.99999999999999963e-105 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y))

   1. Initial program 30.4

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

   2. Simplified 1.2

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}} \]

   if -2e-8 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < -5.00000000000000014e-307 or 0.0 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < 4.99999999999999963e-105

   1. Initial program 0.6

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -5 \cdot 10^{-307}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

  (/ (* (* x 2.0) y) (- x y)))