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

Average Error: 10.0 → 0.7

Time: 6.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.568575037803819 \cdot 10^{+51}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \leq 9.311628134350187 \cdot 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, \frac{x}{z}\right) - x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.568575037803819e+51)
   (- (* (+ y 1.0) (/ x z)) x)
   (if (<= x 9.311628134350187e-149)
     (- (/ (fma x y x) z) x)
     (- (fma x (/ y z) (/ 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 tmp;
	if (x <= -2.568575037803819e+51) {
		tmp = ((y + 1.0) * (x / z)) - x;
	} else if (x <= 9.311628134350187e-149) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = fma(x, (y / z), (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	0.7

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

Derivation

1. Split input into 3 regimes

2. if x < -2.568575037803819e51

   1. Initial program 30.7

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

   2. Simplified 30.7

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

   3. Taylor expanded in y around 0 9.8

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

   4. Simplified 0.0

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

   5. Taylor expanded in x around 0 0.1

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

   6. Applied div-inv_binary64 0.1

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

   7. Applied distribute-lft1-in_binary64 0.1

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

   8. Applied associate-*l*_binary64 0.1

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

   9. Simplified 0.0

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

   if -2.568575037803819e51 < x < 9.31162813435018738e-149

   1. Initial program 0.7

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

   2. Simplified 0.7

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

   3. Taylor expanded in y around 0 0.4

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

   4. Simplified 3.3

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

   5. Taylor expanded in x around 0 5.9

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

   6. Applied *-un-lft-identity_binary64 5.9

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

   7. Applied associate-*l*_binary64 5.9

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

   8. Simplified 0.4

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

   if 9.31162813435018738e-149 < x

   1. Initial program 15.0

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

   2. Simplified 15.0

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

   3. Taylor expanded in y around 0 5.3

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

   4. Simplified 0.2

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

   5. Taylor expanded in x around 0 1.5

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

   6. Applied add-cube-cbrt_binary64 1.7

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}\right) \cdot \sqrt[3]{\frac{y}{z}}} + \frac{1}{z}\right) \cdot x - x \]

   7. Applied fma-def_binary64 1.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}, \sqrt[3]{\frac{y}{z}}, \frac{1}{z}\right)} \cdot x - x \]

   8. Taylor expanded in x around 0 1.5

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

   9. Simplified 1.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.568575037803819 \cdot 10^{+51}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \leq 9.311628134350187 \cdot 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, \frac{x}{z}\right) - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(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))