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

Average Error: 10.7 → 0.1

Time: 2.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ t_1 := \mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\ \mathbf{if}\;t_0 \leq -3.808894655279025 \cdot 10^{+282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 9.734001167037195 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (- (fma (/ x z) y (/ x z)) x)))
   (if (<= t_0 -3.808894655279025e+282)
     t_1
     (if (<= t_0 9.734001167037195e-26) (- (/ (fma x y 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 = fma((x / z), y, (x / z)) - x;
	double tmp;
	if (t_0 <= -3.808894655279025e+282) {
		tmp = t_1;
	} else if (t_0 <= 9.734001167037195e-26) {
		tmp = (fma(x, y, 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(fma(Float64(x / z), y, Float64(x / z)) - x)
	tmp = 0.0
	if (t_0 <= -3.808894655279025e+282)
		tmp = t_1;
	elseif (t_0 <= 9.734001167037195e-26)
		tmp = Float64(Float64(fma(x, y, 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[(x / z), $MachinePrecision] * y + N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -3.808894655279025e+282], t$95$1, If[LessEqual[t$95$0, 9.734001167037195e-26], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.7
Target	0.4
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) < -3.8088946552790251e282 or 9.73400116703719541e-26 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 25.3

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

   2. Simplified 25.3

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

   3. Taylor expanded in y around 0 8.5

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

   4. Simplified 0.1

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

   if -3.8088946552790251e282 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 9.73400116703719541e-26

   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. Simplified 2.5

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

   5. Taylor expanded in x around 0 2.6

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

   6. Taylor expanded in y around 0 0.1

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

   7. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -3.808894655279025 \cdot 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 9.734001167037195 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))