Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.4 → 0.3

Time: 4.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq -845.3025145409939\right) \land \left(t_0 \leq 2.3699023649114968 \cdot 10^{-33} \lor \neg \left(t_0 \leq 2.700355737444232 \cdot 10^{+296}\right)\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (or (<= t_0 (- INFINITY))
           (and (not (<= t_0 -845.3025145409939))
                (or (<= t_0 2.3699023649114968e-33)
                    (not (<= t_0 2.700355737444232e+296)))))
     (fma x (/ y z) x)
     t_0)))

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)) || (!(t_0 <= -845.3025145409939) && ((t_0 <= 2.3699023649114968e-33) || !(t_0 <= 2.700355737444232e+296)))) {
		tmp = fma(x, (y / z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.4
Target	3.3
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -inf.0 or -845.302514540993911 < (/.f64 (*.f64 x (+.f64 y z)) z) < 2.36990236491149677e-33 or 2.7003557374442319e296 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 22.0

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

   2. Simplified 0.3

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -845.302514540993911 or 2.36990236491149677e-33 < (/.f64 (*.f64 x (+.f64 y z)) z) < 2.7003557374442319e296

   1. Initial program 0.2

      \[\frac{x \cdot \left(y + z\right)}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq -845.3025145409939\right) \land \left(\frac{x \cdot \left(y + z\right)}{z} \leq 2.3699023649114968 \cdot 10^{-33} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq 2.700355737444232 \cdot 10^{+296}\right)\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))