Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.4 → 0.3

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x + \frac{1}{\frac{\frac{z}{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := x + \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -2.5470166038654833 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 58567475251.42587:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;t_0 \leq 3.111711601723932 \cdot 10^{+296}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{y}{z} \cdot \sqrt[3]{x}\right)\\ \end{array}\\ \end{array} \]

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))
     (+ x (/ 1.0 (/ (/ z x) y)))
     (let* ((t_1 (+ x (/ (* x y) z))))
       (if (<= t_0 -2.5470166038654833e-217)
         t_1
         (if (<= t_0 58567475251.42587)
           (fma x (/ y z) x)
           (if (<= t_0 3.111711601723932e+296)
             t_1
             (+ x (* (* (cbrt x) (cbrt x)) (* (/ y z) (cbrt 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 = x + (1.0 / ((z / x) / y));
	} else {
		double t_1 = x + ((x * y) / z);
		double tmp_1;
		if (t_0 <= -2.5470166038654833e-217) {
			tmp_1 = t_1;
		} else if (t_0 <= 58567475251.42587) {
			tmp_1 = fma(x, (y / z), x);
		} else if (t_0 <= 3.111711601723932e+296) {
			tmp_1 = t_1;
		} else {
			tmp_1 = x + ((cbrt(x) * cbrt(x)) * ((y / z) * cbrt(x)));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.4
Target	3.2
Herbie	0.3

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

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. Simplified 0.1

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

   3. Taylor expanded in y around 0 20.1

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

   4. Applied associate-/l*_binary64 0.1

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

   5. Applied clear-num_binary64 0.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z}{x}}{y}}} + x \]

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -2.54701660386548331e-217 or 58567475251.425873 < (/.f64 (*.f64 x (+.f64 y z)) z) < 3.111711601723932e296

   1. Initial program 0.3

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

   2. Simplified 5.9

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

   3. Taylor expanded in y around 0 0.2

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

   if -2.54701660386548331e-217 < (/.f64 (*.f64 x (+.f64 y z)) z) < 58567475251.425873

   1. Initial program 9.9

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

   2. Simplified 0.1

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

   if 3.111711601723932e296 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 59.1

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

   2. Simplified 1.4

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

   3. Taylor expanded in y around 0 23.9

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

   4. Applied associate-/l*_binary64 2.7

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

   5. Applied add-cube-cbrt_binary64 2.9

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

   6. Applied *-un-lft-identity_binary64 2.9

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

   7. Applied times-frac_binary64 2.9

      \[\leadsto \frac{y}{\color{blue}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{z}{\sqrt[3]{x}}}} + x \]

   8. Applied *-un-lft-identity_binary64 2.9

      \[\leadsto \frac{\color{blue}{1 \cdot y}}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{z}{\sqrt[3]{x}}} + x \]

   9. Applied times-frac_binary64 2.0

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{y}{\frac{z}{\sqrt[3]{x}}}} + x \]

   10. Simplified 2.0

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

   11. Simplified 1.6

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;x + \frac{1}{\frac{\frac{z}{x}}{y}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -2.5470166038654833 \cdot 10^{-217}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 58567475251.42587:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 3.111711601723932 \cdot 10^{+296}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{y}{z} \cdot \sqrt[3]{x}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022097 
(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))