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

Average Error: 10.1 → 0.1

Time: 2.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ t_1 := \left(y + 1\right) \cdot \frac{x}{z} - x\\ \mathbf{if}\;t_0 \leq -5.697622636079628 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 7.231401267720252 \cdot 10^{+196}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (- y z) 1.0)) z)) (t_1 (- (* (+ y 1.0) (/ x z)) x)))
   (if (<= t_0 -5.697622636079628e+53)
     t_1
     (if (<= t_0 7.231401267720252e+196) (- (/ (fma y x x) z) x) t_1))))

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 * ((y - z) + 1.0)) / z;
	double t_1 = ((y + 1.0) * (x / z)) - x;
	double tmp;
	if (t_0 <= -5.697622636079628e+53) {
		tmp = t_1;
	} else if (t_0 <= 7.231401267720252e+196) {
		tmp = (fma(y, x, x) / z) - x;
	} else {
		tmp = t_1;
	}
	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(Float64(y - z) + 1.0)) / z)
	t_1 = Float64(Float64(Float64(y + 1.0) * Float64(x / z)) - x)
	tmp = 0.0
	if (t_0 <= -5.697622636079628e+53)
		tmp = t_1;
	elseif (t_0 <= 7.231401267720252e+196)
		tmp = Float64(Float64(fma(y, x, x) / z) - x);
	else
		tmp = t_1;
	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[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y + 1.0), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -5.697622636079628e+53], t$95$1, If[LessEqual[t$95$0, 7.231401267720252e+196], N[(N[(N[(y * x + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.1
Target	0.5
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -5.69762263607962818e53 or 7.23140126772025163e196 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 25.0

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

   2. Simplified 25.0

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

   3. Taylor expanded in y around 0 7.9

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

   4. Applied egg-rr 0.1

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

   5. Applied egg-rr 0.1

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

   if -5.69762263607962818e53 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 7.23140126772025163e196

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   3. Taylor expanded in y around 0 0.1

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

   4. Applied egg-rr 0.1

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

   5. Taylor expanded in z around 0 0.1

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

   6. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -5.697622636079628 \cdot 10^{+53}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 7.231401267720252 \cdot 10^{+196}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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))