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

Average Error: 10.0 → 0.1

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y + 1}{z} - x\\ \mathbf{elif}\;t_0 \leq 1.1652699375053744 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\ \end{array} \]

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)))
   (if (<= t_0 (- INFINITY))
     (- (* x (/ (+ y 1.0) z)) x)
     (if (<= t_0 1.1652699375053744e-20)
       (- (/ (fma x y x) z) x)
       (- (* (+ y 1.0) (/ x z)) x)))))

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 tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * ((y + 1.0) / z)) - x;
	} else if (t_0 <= 1.1652699375053744e-20) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = ((y + 1.0) * (x / z)) - x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.0
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 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded in y around 0 21.6

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

   4. Simplified 0.1

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

   5. Taylor expanded in x around 0 0.0

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

   6. Taylor expanded in z around 0 0.0

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

   7. Simplified 0.0

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 1.16526993750537441e-20

   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 2.6

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

   5. Taylor expanded in x around 0 2.8

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

   6. Applied pow1_binary64 2.8

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

   7. Applied pow1_binary64 2.8

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

   8. Applied pow-prod-down_binary64 2.8

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

   9. Simplified 0.1

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

   if 1.16526993750537441e-20 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 15.4

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

   2. Simplified 15.4

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

   3. Taylor expanded in y around 0 4.8

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

   4. Simplified 0.1

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

   5. Taylor expanded in x around 0 6.2

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

   6. Applied div-inv_binary64 6.3

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

   7. Applied distribute-lft1-in_binary64 6.3

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

   8. Applied associate-*l*_binary64 0.2

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

   9. Simplified 0.1

      \[\leadsto \left(y + 1\right) \cdot \color{blue}{\frac{x}{z}} - x \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -\infty:\\ \;\;\;\;x \cdot \frac{y + 1}{z} - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 1.1652699375053744 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022066 
(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))