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

Average Error: 4.3 → 2.4

Time: 11.6s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \le +inf.0:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \log \left(e^{\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}} \cdot \sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) ((((double) (z * ((double) sqrt(((double) (t + a)))))) / t) - ((double) (((double) (b - c)) * ((double) (((double) (a + (5.0 / 6.0))) - (2.0 / ((double) (t * 3.0))))))))))))))))));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double VAR;
	if ((((double) ((((double) (z * ((double) sqrt(((double) (t + a)))))) / t) - ((double) (((double) (b - c)) * ((double) (((double) (a + (5.0 / 6.0))) - (2.0 / ((double) (t * 3.0))))))))) <= +inf.0)) {
		VAR = (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) log(((double) exp(((double) ((((double) (z * ((double) sqrt(((double) (t + a)))))) / t) - ((double) (((double) (b - c)) * ((double) (((double) (a + (5.0 / 6.0))) - (2.0 / ((double) (t * 3.0))))))))))))))))))))));
	} else {
		VAR = ((double) ((((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / ((double) (((double) cbrt(((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) (((double) (c * ((double) (a + 0.8333333333333334)))) - ((double) (a * b)))))))))))))) * ((double) cbrt(((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) (((double) (c * ((double) (a + 0.8333333333333334)))) - ((double) (a * b))))))))))))))))) * (((double) cbrt(x)) / ((double) cbrt(((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) (((double) (c * ((double) (a + 0.8333333333333334)))) - ((double) (a * b)))))))))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	4.3
Target	3.2
Herbie	2.4

\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))) < +inf.0

   1. Initial program 0.8

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied add-log-exp 5.4

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\log \left(e^{\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)}\right)}\right)}}\]

   4. Applied add-log-exp 14.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\log \left(e^{\frac{z \cdot \sqrt{t + a}}{t}}\right)} - \log \left(e^{\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)}\right)\right)}}\]

   5. Applied diff-log 14.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\log \left(\frac{e^{\frac{z \cdot \sqrt{t + a}}{t}}}{e^{\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)}}\right)}}}\]

   6. Simplified 0.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \log \color{blue}{\left(e^{\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)}\right)}}}\]

   if +inf.0 < (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))

   1. Initial program 64.0

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   2. Taylor expanded around inf 29.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}}\]

   3. Simplified 29.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 29.4

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}} \cdot \sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}\right) \cdot \sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}}}\]

   6. Applied add-cube-cbrt 29.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}} \cdot \sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}\right) \cdot \sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}}\]

   7. Applied times-frac 29.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}} \cdot \sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \le +inf.0:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \log \left(e^{\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}} \cdot \sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (* t t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))