Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.6 → 0.2

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{elif}\;t_0 \leq -6.0939414944138854 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 3.3002802206311174 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;t_0 \leq 8.894138492345838 \cdot 10^{+298}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ y z)) z)))
   (if (<= t_0 (- INFINITY))
     (fma (/ x z) y x)
     (if (<= t_0 -6.0939414944138854e+26)
       t_0
       (if (<= t_0 3.3002802206311174e+30)
         (/ x (/ z (+ y z)))
         (if (<= t_0 8.894138492345838e+298) t_0 (fma x (/ y z) x)))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((x / z), y, x);
	} else if (t_0 <= -6.0939414944138854e+26) {
		tmp = t_0;
	} else if (t_0 <= 3.3002802206311174e+30) {
		tmp = x / (z / (y + z));
	} else if (t_0 <= 8.894138492345838e+298) {
		tmp = t_0;
	} else {
		tmp = fma(x, (y / z), x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.6
Target	3.2
Herbie	0.2

\[\frac{x}{\frac{z}{y + z}} \]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Taylor expanded in y around 0 21.2

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

   3. Simplified 0.0

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -6.09394149441388545e26 or 3.30028022063111741e30 < (/.f64 (*.f64 x (+.f64 y z)) z) < 8.8941384923458379e298

   1. Initial program 0.2

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

   if -6.09394149441388545e26 < (/.f64 (*.f64 x (+.f64 y z)) z) < 3.30028022063111741e30

   1. Initial program 5.8

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

   2. Applied associate-/l*_binary64 0.1

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

   if 8.8941384923458379e298 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 60.3

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

   2. Simplified 0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -6.0939414944138854 \cdot 10^{+26}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 3.3002802206311174 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 8.894138492345838 \cdot 10^{+298}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

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