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

Average Error: 4.4 → 0.5

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1.793197774245199 \cdot 10^{-184}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - \frac{t}{1 - z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 3.575645117889444 \cdot 10^{-238}:\\ \;\;\;\;\frac{x}{z} \cdot \left(t + \left(y + \frac{t}{z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2.6004804340552325 \cdot 10^{+205}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - \frac{t}{1 - z} \cdot x\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (/ y z) (/ t (- 1.0 z))) (- INFINITY))
   (/ (* x (- y (* z (+ y t)))) (* z (- 1.0 z)))
   (if (<= (- (/ y z) (/ t (- 1.0 z))) -1.793197774245199e-184)
     (- (/ x (/ z y)) (* (/ t (- 1.0 z)) x))
     (if (<= (- (/ y z) (/ t (- 1.0 z))) 3.575645117889444e-238)
       (* (/ x z) (+ t (+ y (/ t z))))
       (if (<= (- (/ y z) (/ t (- 1.0 z))) 2.6004804340552325e+205)
         (- (* (/ y z) x) (* (/ t (- 1.0 z)) x))
         (- (* y (/ x z)) (* (/ t (- 1.0 z)) x)))))))

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 tmp;
	if (((y / z) - (t / (1.0 - z))) <= -((double) INFINITY)) {
		tmp = (x * (y - (z * (y + t)))) / (z * (1.0 - z));
	} else if (((y / z) - (t / (1.0 - z))) <= -1.793197774245199e-184) {
		tmp = (x / (z / y)) - ((t / (1.0 - z)) * x);
	} else if (((y / z) - (t / (1.0 - z))) <= 3.575645117889444e-238) {
		tmp = (x / z) * (t + (y + (t / z)));
	} else if (((y / z) - (t / (1.0 - z))) <= 2.6004804340552325e+205) {
		tmp = ((y / z) * x) - ((t / (1.0 - z)) * x);
	} else {
		tmp = (y * (x / z)) - ((t / (1.0 - z)) * x);
	}
	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	0.5

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

Derivation

1. Split input into 5 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -inf.0

   1. Initial program 64.0

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied frac-sub_binary64_9859 64.0

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

   4. Applied associate-*r/_binary64_9794 0.3

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

   5. Simplified 0.3

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -1.7931977742451991e-184

   1. Initial program 0.2

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied sub-neg_binary64_9843 0.2

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

   4. Applied distribute-rgt-in_binary64_9802 0.2

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

   5. Simplified 0.2

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

   6. Simplified 0.2

      \[\leadsto x \cdot \frac{y}{z} + \color{blue}{x \cdot \left(-\frac{t}{1 - z}\right)}\]
   7. Using strategy rm

   8. Applied associate-*r/_binary64_9794 6.1

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
   9. Using strategy rm

   10. Applied associate-/l*_binary64_9797 0.2

       \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
   11. Using strategy rm

   12. Applied distribute-rgt-neg-out_binary64_9812 0.2

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

   13. Applied unsub-neg_binary64_9844 0.2

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

   if -1.7931977742451991e-184 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 3.57564511788944403e-238

   1. Initial program 9.4

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

   2. Taylor expanded around inf 1.9

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} + \left(\frac{t \cdot x}{z} + \frac{t \cdot x}{{z}^{2}}\right)}\]

   3. Simplified 2.0

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

   if 3.57564511788944403e-238 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.60048043405523248e205

   1. Initial program 0.2

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied sub-neg_binary64_9843 0.2

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

   4. Applied distribute-rgt-in_binary64_9802 0.2

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

   5. Simplified 0.2

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

   6. Simplified 0.2

      \[\leadsto x \cdot \frac{y}{z} + \color{blue}{x \cdot \left(-\frac{t}{1 - z}\right)}\]
   7. Using strategy rm

   8. Applied distribute-rgt-neg-out_binary64_9812 0.2

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

   9. Applied unsub-neg_binary64_9844 0.2

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

   if 2.60048043405523248e205 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 18.0

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied sub-neg_binary64_9843 18.0

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

   4. Applied distribute-rgt-in_binary64_9802 18.0

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

   5. Simplified 18.0

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

   6. Simplified 18.0

      \[\leadsto x \cdot \frac{y}{z} + \color{blue}{x \cdot \left(-\frac{t}{1 - z}\right)}\]
   7. Using strategy rm

   8. Applied associate-*r/_binary64_9794 1.2

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
   9. Using strategy rm

   10. Applied associate-/l*_binary64_9797 16.6

       \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
   11. Using strategy rm

   12. Applied associate-/r/_binary64_9798 1.2

       \[\leadsto \color{blue}{\frac{x}{z} \cdot y} + x \cdot \left(-\frac{t}{1 - z}\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1.793197774245199 \cdot 10^{-184}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - \frac{t}{1 - z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 3.575645117889444 \cdot 10^{-238}:\\ \;\;\;\;\frac{x}{z} \cdot \left(t + \left(y + \frac{t}{z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2.6004804340552325 \cdot 10^{+205}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - \frac{t}{1 - z} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(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)))))