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

Average Error: 4.6 → 1.3

Time: 4.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -6.28902280085759557 \cdot 10^{184}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.2136975206029316 \cdot 10^{-125}:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(\sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}} \cdot \sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}}\right) \cdot \sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.9453220644851443 \cdot 10^{-279}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 6.56399033386167465 \cdot 10^{218}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z}\right) + \left(-x\right) \cdot \frac{t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\\ \end{array}\]

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 temp;
	if ((((y / z) - (t / (1.0 - z))) <= -6.2890228008575956e+184)) {
		temp = (((x * y) * (1.0 / z)) + (-x * (t / (1.0 - z))));
	} else {
		double temp_1;
		if ((((y / z) - (t / (1.0 - z))) <= -1.2136975206029316e-125)) {
			temp_1 = ((x * (y / z)) + ((cbrt((-x * (t / (1.0 - z)))) * cbrt((-x * (t / (1.0 - z))))) * cbrt((-x * (t / (1.0 - z))))));
		} else {
			double temp_2;
			if ((((y / z) - (t / (1.0 - z))) <= 2.9453220644851443e-279)) {
				temp_2 = (((x * y) / z) + ((1.0 * ((t * x) / pow(z, 2.0))) + ((t * x) / z)));
			} else {
				double temp_3;
				if ((((y / z) - (t / (1.0 - z))) <= 6.563990333861675e+218)) {
					temp_3 = (((cbrt(x) * cbrt(x)) * (cbrt(x) * (y / z))) + (-x * (t / (1.0 - z))));
				} else {
					temp_3 = (((x * y) * (1.0 / z)) + (-x * (t / (1.0 - z))));
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.6
Target	4.2
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -6.2890228008575956e+184 or 6.563990333861675e+218 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 19.3

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

   3. Applied div-inv 19.3

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

   5. Applied sub-neg 19.3

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

   6. Applied distribute-lft-in 19.3

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

   7. Simplified 19.3

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

   9. Applied div-inv 19.3

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

   10. Applied associate-*r* 0.9

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

   if -6.2890228008575956e+184 < (- (/ y z) (/ t (- 1.0 z))) < -1.2136975206029316e-125

   1. Initial program 0.2

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

   3. Applied div-inv 0.2

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

   5. Applied sub-neg 0.2

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

   6. Applied distribute-lft-in 0.2

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

   7. Simplified 0.2

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

   9. Applied add-cube-cbrt 0.7

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

   if -1.2136975206029316e-125 < (- (/ y z) (/ t (- 1.0 z))) < 2.9453220644851443e-279

   1. Initial program 7.9

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

   3. Applied div-inv 8.0

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

   4. Taylor expanded around inf 4.2

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)}\]

   if 2.9453220644851443e-279 < (- (/ y z) (/ t (- 1.0 z))) < 6.563990333861675e+218

   1. Initial program 0.2

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

   3. Applied div-inv 0.2

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

   5. Applied sub-neg 0.2

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

   6. Applied distribute-lft-in 0.2

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

   7. Simplified 0.2

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

   9. Applied add-cube-cbrt 0.8

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

   10. Applied associate-*l* 0.8

       \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z}\right)} + \left(-x\right) \cdot \frac{t}{1 - z}\]
3. Recombined 4 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -6.28902280085759557 \cdot 10^{184}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.2136975206029316 \cdot 10^{-125}:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(\sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}} \cdot \sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}}\right) \cdot \sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.9453220644851443 \cdot 10^{-279}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 6.56399033386167465 \cdot 10^{218}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z}\right) + \left(-x\right) \cdot \frac{t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

  (* x (- (/ y z) (/ t (- 1 z)))))