Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C

Average Error: 4.4 → 1.4

Time: 12.9s

Precision: binary64

Math

\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]

↓

\[\begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := \frac{t \cdot x}{1 - z}\\ \mathbf{if}\;t_1 \leq -1.1584484572454943 \cdot 10^{+250}:\\ \;\;\;\;\frac{y \cdot x}{z} - t_2\\ \mathbf{elif}\;t_1 \leq 5.659525696432036 \cdot 10^{+181}:\\ \;\;\;\;t_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, -t_2\right)\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))) (t_2 (/ (* t x) (- 1.0 z))))
   (if (<= t_1 -1.1584484572454943e+250)
     (- (/ (* y x) z) t_2)
     (if (<= t_1 5.659525696432036e+181) (* t_1 x) (fma y (/ x z) (- t_2))))))

C

double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double t_2 = (t * x) / (1.0 - z);
	double tmp;
	if (t_1 <= -1.1584484572454943e+250) {
		tmp = ((y * x) / z) - t_2;
	} else if (t_1 <= 5.659525696432036e+181) {
		tmp = t_1 * x;
	} else {
		tmp = fma(y, (x / z), -t_2);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.4
Target	4.1
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -1.1584484572454943e250

   1. Initial program 27.9

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]

   2. Taylor expanded in y around 0 0.3

      \[\leadsto \color{blue}{\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}} \]

   if -1.1584484572454943e250 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 5.6595256964320361e181

   1. Initial program 1.5

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]

   2. Taylor expanded in y around 0 8.2

      \[\leadsto \color{blue}{\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}} \]

   3. Applied add-cube-cbrt_binary64 8.6

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

   4. Applied add-cube-cbrt_binary64 9.1

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

   5. Applied prod-diff_binary64 9.1

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

   6. Simplified 5.2

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

   7. Simplified 1.5

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) + \color{blue}{0} \]

   if 5.6595256964320361e181 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 17.1

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]

   2. Taylor expanded in y around 0 1.4

      \[\leadsto \color{blue}{\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}} \]

   3. Applied *-un-lft-identity_binary64 1.4

      \[\leadsto \frac{y \cdot x}{\color{blue}{1 \cdot z}} - \frac{t \cdot x}{1 - z} \]

   4. Applied times-frac_binary64 1.9

      \[\leadsto \color{blue}{\frac{y}{1} \cdot \frac{x}{z}} - \frac{t \cdot x}{1 - z} \]

   5. Applied fma-neg_binary64 1.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{1}, \frac{x}{z}, -\frac{t \cdot x}{1 - z}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1.1584484572454943 \cdot 10^{+250}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 5.659525696432036 \cdot 10^{+181}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, -\frac{t \cdot x}{1 - z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022117 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))