Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.5 → 0.7

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ t_1 := \frac{x}{\frac{z}{y + z}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1.9758423335381378 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 1.462732982746968 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 1.6895306086943411 \cdot 10^{+249}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ y z)) z)) (t_1 (/ x (/ z (+ y z)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -1.9758423335381378e+60)
       t_0
       (if (<= t_0 1.462732982746968e-41)
         t_1
         (if (<= t_0 1.6895306086943411e+249) 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 t_1 = x / (z / (y + z));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -1.9758423335381378e+60) {
		tmp = t_0;
	} else if (t_0 <= 1.462732982746968e-41) {
		tmp = t_1;
	} else if (t_0 <= 1.6895306086943411e+249) {
		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	12.5
Target	2.9
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -inf.0 or -1.9758423335381378e60 < (/.f64 (*.f64 x (+.f64 y z)) z) < 1.4627329827469679e-41

   1. Initial program 15.0

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Using strategy rm

   3. Applied associate-/l*_binary64 0.2

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -1.9758423335381378e60 or 1.4627329827469679e-41 < (/.f64 (*.f64 x (+.f64 y z)) z) < 1.6895306086943411e249

   1. Initial program 0.2

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

   if 1.6895306086943411e249 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 43.3

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

   2. Simplified 4.5

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -1.9758423335381378 \cdot 10^{+60}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 1.462732982746968 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 1.6895306086943411 \cdot 10^{+249}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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))