Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Average Error: 9.9 → 0.2

Time: 2.9s

Precision: binary64

Math

\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

↓

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}} - x\\ \mathbf{if}\;z \leq -1 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 16000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ x (/ z y)) x)))
   (if (<= z -1e+73) t_0 (if (<= z 16000000.0) (- (/ (fma x y x) z) x) t_0))))

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

↓

double code(double x, double y, double z) {
	double t_0 = (x / (z / y)) - x;
	double tmp;
	if (z <= -1e+73) {
		tmp = t_0;
	} else if (z <= 16000000.0) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(x / Float64(z / y)) - x)
	tmp = 0.0
	if (z <= -1e+73)
		tmp = t_0;
	elseif (z <= 16000000.0)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[z, -1e+73], t$95$0, If[LessEqual[z, 16000000.0], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	9.9
Target	0.5
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999983e72 or 1.6e7 < z

   1. Initial program 18.1

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Simplified 5.7

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

   3. Applied egg-rr 5.7

      \[\leadsto \color{blue}{{\left(\frac{z}{\mathsf{fma}\left(x, y, x\right)}\right)}^{-1}} - x \]

   4. Taylor expanded in y around inf 5.8

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

   5. Simplified 0.2

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

   if -9.99999999999999983e72 < z < 1.6e7

   1. Initial program 0.6

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

   2. Simplified 0.3

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+73}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{elif}\;z \leq 16000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))