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

Average Error: 4.6 → 0.8

Time: 12.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y \cdot \left(1 - z\right) - z \cdot t}{1 - z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq -9.213416733986507 \cdot 10^{-195}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 0:\\ \;\;\;\;\frac{x \cdot y}{z} - \frac{x \cdot t}{1 - z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 1.5315100100377353 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \frac{y}{z} - x \cdot \frac{t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x (- (/ y z) (/ t (- 1.0 z)))) (- INFINITY))
   (* (/ x z) (/ (- (* y (- 1.0 z)) (* z t)) (- 1.0 z)))
   (if (<= (* x (- (/ y z) (/ t (- 1.0 z)))) -9.213416733986507e-195)
     (* x (- (/ y z) (/ t (- 1.0 z))))
     (if (<= (* x (- (/ y z) (/ t (- 1.0 z)))) 0.0)
       (- (/ (* x y) z) (/ (* x t) (- 1.0 z)))
       (if (<= (* x (- (/ y z) (/ t (- 1.0 z)))) 1.5315100100377353e+295)
         (- (* x (/ y z)) (* x (/ t (- 1.0 z))))
         (/ (* x (- (* y (- 1.0 z)) (* z t))) (* z (- 1.0 z))))))))

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 ((x * ((y / z) - (t / (1.0 - z)))) <= -((double) INFINITY)) {
		tmp = (x / z) * (((y * (1.0 - z)) - (z * t)) / (1.0 - z));
	} else if ((x * ((y / z) - (t / (1.0 - z)))) <= -9.213416733986507e-195) {
		tmp = x * ((y / z) - (t / (1.0 - z)));
	} else if ((x * ((y / z) - (t / (1.0 - z)))) <= 0.0) {
		tmp = ((x * y) / z) - ((x * t) / (1.0 - z));
	} else if ((x * ((y / z) - (t / (1.0 - z)))) <= 1.5315100100377353e+295) {
		tmp = (x * (y / z)) - (x * (t / (1.0 - z)));
	} else {
		tmp = (x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.6
Target	4.3
Herbie	0.8

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

Derivation

1. Split input into 5 regimes

2. if (*.f64 x (-.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 add-cube-cbrt_binary64_10002 64.0

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

   4. Applied associate-*l*_binary64_9908 64.0

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

   5. Simplified 64.0

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

   7. Applied frac-sub_binary64_9976 64.0

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

   8. Applied associate-*l/_binary64_9910 36.2

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

   9. Applied associate-*r/_binary64_9909 1.5

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

   10. Simplified 0.3

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

   12. Applied times-frac_binary64_9973 0.2

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

   if -inf.0 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < -9.21341673398650663e-195

   1. Initial program 0.3

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

   if -9.21341673398650663e-195 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < 0.0

   1. Initial program 6.2

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

   3. Applied add-cube-cbrt_binary64_10002 6.5

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

   4. Applied associate-*l*_binary64_9908 6.5

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

   5. Simplified 6.5

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

   6. Taylor expanded around 0 2.4

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

   if 0.0 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < 1.5315100100377353e295

   1. Initial program 0.3

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

   3. Applied sub-neg_binary64_9960 0.3

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

   4. Applied distribute-rgt-in_binary64_9917 0.3

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

   if 1.5315100100377353e295 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))))

   1. Initial program 50.4

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

   3. Applied add-cube-cbrt_binary64_10002 50.7

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

   4. Applied associate-*l*_binary64_9908 50.7

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

   5. Simplified 50.7

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

   7. Applied frac-sub_binary64_9976 52.5

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

   8. Applied associate-*l/_binary64_9910 32.7

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

   9. Applied associate-*r/_binary64_9909 6.1

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

   10. Simplified 5.0

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y \cdot \left(1 - z\right) - z \cdot t}{1 - z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq -9.213416733986507 \cdot 10^{-195}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 0:\\ \;\;\;\;\frac{x \cdot y}{z} - \frac{x \cdot t}{1 - z}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 1.5315100100377353 \cdot 10^{+295}:\\ \;\;\;\;x \cdot \frac{y}{z} - x \cdot \frac{t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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)))))