Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.1 → 0.4

Time: 2.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(y + z\right)\\ t_1 := \frac{t_0}{z}\\ t_2 := \mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -6.928840578777105 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 5.5452176622010715 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2.7293408556137423 \cdot 10^{+294}:\\ \;\;\;\;t_0 \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ y z))) (t_1 (/ t_0 z)) (t_2 (fma x (/ y z) x)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -6.928840578777105e+20)
       t_1
       (if (<= t_1 5.5452176622010715e+41)
         t_2
         (if (<= t_1 2.7293408556137423e+294)
           (* t_0 (/ 1.0 z))
           (fma (/ x z) y 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);
	double t_1 = t_0 / z;
	double t_2 = fma(x, (y / z), x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -6.928840578777105e+20) {
		tmp = t_1;
	} else if (t_1 <= 5.5452176622010715e+41) {
		tmp = t_2;
	} else if (t_1 <= 2.7293408556137423e+294) {
		tmp = t_0 * (1.0 / z);
	} else {
		tmp = fma((x / z), y, x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.1
Target	2.9
Herbie	0.4

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -inf.0 or -692884057877710504000 < (/.f64 (*.f64 x (+.f64 y z)) z) < 5.54521766220107147e41

   1. Initial program 13.4

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

   2. Simplified 0.2

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -692884057877710504000

   1. Initial program 0.2

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

   if 5.54521766220107147e41 < (/.f64 (*.f64 x (+.f64 y z)) z) < 2.72934085561374228e294

   1. Initial program 0.2

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

   2. Applied div-inv_binary64 0.3

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

   if 2.72934085561374228e294 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 59.5

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

   2. Taylor expanded in y around 0 21.2

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

   3. Simplified 2.7

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -6.928840578777105 \cdot 10^{+20}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 5.5452176622010715 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 2.7293408556137423 \cdot 10^{+294}:\\ \;\;\;\;\left(x \cdot \left(y + z\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))